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In: Markakis, E., Schäfer, G. (eds.) Web and Internet Economics. Lecture Notes in Computer Science, vol. 9470. Springer, Berlin (2015)\nBerger, E.: Dynamic monopolies of constant size. J. Comb. Theory Ser. B 83(2), 191–200 (2001)\nBroere, I., Hattingh, J., Henning, M.A., McRae, A.: Majority domination in graphs. Discrete Mathematics 138(1), 125–135 (1995)\nCaro, Y., Hansberg, A., Montejano, A.: Zero-sum subsequences in bounded-sum \\(\\{-1,1\\}\\)-sequences (2016). arXiv:1612.06523\nCurtis, E., Ingerman, D., Morrow, J.: Circular planar graphs and resistor networks. Linear Algebra Appl. 283(1–3), 115–150 (1998)\nErdős, P.: Graph theory and probability. Can. J. Math. 11, 34–38 (1959)\nFishburn, P., Hwang, F., Lee, H.: Do local majorities force a global majority? Discrete Math. 61(2), 165–179 (1986)\nFüredi, Z., Mubayi, D.: Signed domination in regular graphs and set-systems. J. Comb. Theory Ser. B 76(2), 223–239 (1999)\nPeleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231–257 (2002)\nPoghosyan, A., Zverovich, V.: Discrepancy and signed domination in graphs and hypergraphs. Discrete Math. 310(15), 2091–2099 (2010)\nTruemper, K.: On the delta-wye reduction for planar graphs. J. Graph Theory 13(2), 141–148 (1989)\nWoodall, D.: Local and global proportionality. Discrete Math. 102(3), 315–328 (1992)\nXu, B.: On signed edge domination numbers of graphs. Discrete Math. 239(1–3), 179–189 (2001)",{"EN":227},"Let \n                  \n                    \n                  \n                  $$\\mathcal G$$\n                  \n                    \n                  \n                 be an infinite family of connected graphs and let k be a positive integer. We say that k is forcing for \n                  \n                    \n                  \n                  $$\\mathcal G$$\n                  \n                    \n                  \n                 if for all \n                  \n                    \n                  \n                  $$G \\in \\mathcal G$$\n                  \n                    \n                  \n                 but finitely many, the following holds. Any \n                  \n                    \n                  \n                  $$\\{-1,1\\}$$\n                  \n                    \n                  \n                -weighing of the edges of G for which all connected subgraphs on k edges are positively weighted implies that G is positively weighted. Otherwise, we say that it is weakly forcing for \n                  \n                    \n                  \n                  $$\\mathcal G$$\n                  \n                    \n                  \n                 if any such weighing implies that the weight of G is bounded from below by a constant. Otherwise we say that k collapses for \n                  \n                    \n                  \n                  $$\\mathcal G$$\n                  \n                    \n                  \n                . We classify k for some of the most prominent classes of graphs, such as all connected graphs, all connected graphs with a given maximum degree and all connected graphs with a given average degree.",{"EN":229},"The Effect of Local Majority on Global Majorityin Connected Graphs",{"VOID":231},"10.1007\u002Fs00373-018-1938-0","PUBLICATION","VERIFIED","Auto 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Caro",{"id":255,"sortIndex":256,"researcher":20,"roles":257,"affiliations":258,"properties":269},"69319b3b-dd9a-4dfd-bd7a-506b59c05f79",1,[240],[259],{"id":20,"sortIndex":21,"affiliation":260,"properties":20},{"id":261,"createTime":262,"updateTime":263,"relativeEntities":264,"slug":265,"properties":266,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"179bdb11-b223-414d-aa89-2c78e2ea5731","2024-01-03T14:28:26.248+00:00","2024-08-31T12:20:58.055+00:00",[],"Department-of-Mathematics-University-of-Haifa-Haifa-Israel",{"title":267},{"VI":268},"Department of Mathematics, University of Haifa, Haifa, Israel",{"title":270},{"VI":271},"Raphael Yuster","ARTICLE",{"url":235,"publisher":274,"properties":302},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":275,"slug":10,"properties":276,"entityType":18,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"subjectFields":280,"manageAffiliations":281,"indexDatabases":282,"url":20,"thumbnailPath":20,"statistic":297,"gsStatistic":20,"type":20,"analyzePriority":20},[],{"issn":277,"eissn":278,"title":279},{"VOID":13},{"VOID":15},{"EN":17},[],[],[283,290],{"id":66,"indexDatabase":284,"url":79,"indexYears":80,"academicFieldIds":289,"indexDatabaseRanking":84},{"id":68,"createTime":69,"updateTime":70,"relativeEntities":285,"label":286,"description":287,"key":76,"publicationTags":288,"standard":20},[],{"EN":73,"VI":73},{"EN":73,"VI":75},[78],[82,83],{"id":86,"indexDatabase":291,"url":101,"indexYears":20,"academicFieldIds":296,"indexDatabaseRanking":20},{"id":88,"createTime":89,"updateTime":90,"relativeEntities":292,"label":293,"description":294,"key":97,"publicationTags":295,"standard":20},[],{"EN":93,"VI":93},{"VI":95,"EN":96},[99,100],[103],{"impactFactor":21,"impactFactorByYear":298,"i10Index":114,"i10IndexLast5Year":115,"totalPublication":116,"totalPublicationByYear":299,"totalCitation":152,"totalCitationByYear":300,"totalCitationPerPublication":176,"totalCitationPerPublicationByYear":301,"hindexLast5Year":120,"hindex":120},{"2012":106,"2013":107,"2014":108,"2015":109,"2016":110,"2017":107,"2018":111,"2019":111,"2020":112,"2021":113,"2022":111,"2023":106},{"1985":118,"1986":119,"1987":120,"1988":121,"1989":122,"1990":123,"1991":124,"1992":125,"1993":126,"1994":127,"1995":124,"1996":128,"1997":129,"1998":124,"1999":130,"2000":120,"2001":131,"2002":132,"2003":133,"2004":127,"2005":134,"2006":130,"2007":135,"2008":136,"2009":137,"2010":138,"2011":139,"2012":140,"2013":141,"2014":142,"2015":143,"2016":144,"2017":145,"2018":146,"2019":147,"2020":148,"2021":149,"2022":150,"2023":151,"2024":127},{"1985":142,"1986":154,"1987":155,"1988":156,"1989":157,"1990":158,"1991":159,"1992":160,"1993":115,"1994":161,"1995":162,"1996":163,"1997":126,"1998":164,"1999":119,"2000":146,"2001":133,"2002":150,"2003":165,"2004":166,"2005":146,"2006":167,"2007":168,"2008":130,"2009":169,"2010":170,"2011":147,"2012":171,"2013":172,"2014":173,"2015":174,"2016":139,"2017":135,"2018":175,"2019":158,"2020":123,"2021":118,"2022":136,"2023":115},{"1985":178,"1986":179,"1987":180,"1988":181,"1989":182,"1990":183,"1991":184,"1992":185,"1993":186,"1994":187,"1995":188,"1996":189,"1997":190,"1998":191,"1999":192,"2000":193,"2001":194,"2002":195,"2003":196,"2004":197,"2005":198,"2006":193,"2007":199,"2008":200,"2009":201,"2010":202,"2011":203,"2012":204,"2013":205,"2014":206,"2015":207,"2016":208,"2017":179,"2018":209,"2019":204,"2020":210,"2021":209,"2022":211,"2023":212},{"volume":303,"pages":305},{"VOID":304},"34",{"VOID":306},"1469-1487","2018-08-28",2018,false,{"id":311,"createTime":312,"updateTime":313,"relativeEntities":314,"slug":315,"properties":316,"entityType":232,"verifyStatus":233,"verifyTime":313,"verifyNote":234,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":325,"fullTextUrl":20,"authors":326,"publicationType":272,"publisherRelationship":357,"citationCount":20,"citationInfo":20,"publishDate":390,"publishYear":308,"citationAnalyzeStatus":19,"lastCitationAnalyze":20,"indexDatabases":20,"openAccess":20,"references":20,"isForceReanalyzing":309},"bb0460bf-c842-42cb-931d-feddc9d0ec2b","2024-01-20T12:33:24.448+00:00","2024-12-31T23:58:23.064+00:00",[],"Graph-Bases-and-Diagram-Commutativity",{"references":317,"abstract":319,"title":321,"doi":323},{"VOID":318},"Diestel, R.: Graph Theory, Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Berlin (2005)\nDixon, E.T., Goodman, S.E.: An algorithm for the longest cycle problem. Networks 6, 139–146 (1976)\nEppstein, D.: StackExchange, April 20, 2015 (on-line discussion). http:\u002F\u002Fcstheory.stackexchange.com\u002Fquestions\u002F31203\u002Fwhat-is-the-best-way-to-find-an-induced-cycle-basis-of-a-graph\nGalluccio, A., Loebl, M.: \\((p, q)\\)-odd digraphs. J. Graph Theory 23(2), 175–184 (1996)\nHammack, R.H., Kainen, P.C.: Robust cycle bases do not exist for \\(K_{n, n}\\) if \\(n \\ge 8\\). Discrete Appl. Math. 235, 206–211 (2018)\nHarary, F.: Graph Theory. Addison-Wesley, Reading (1969)\nKainen, P.C.: On robust cycle bases. Electron. Notes Discret. Math. 11, 430–437 (2002) [Proc. 9th Quadr. Conf. on Graph Theory, Comb., Algorithms and Appl., ed. by Y. Alavi et al., 2000)]\nKainen, P.C.: Isolated squares in hypercubes and robustness of commutativity. Cahiers de topologie et géom. diff. catég 43(3), 213–220 (2002)\nKainen, P.C.: Graph cycles and diagram commutativity. Diagrammes 67–68, 177–238 (2012) (Supplem.)\nKainen, P.C.: Cycle construction and geodesic cycles with application to the hypercube. Ars Math. Contemp. 9(1), 27–43 (2015)\nKlemm, K., Stadler, P.F.: Statistics of cycles in large networks. Phys. Rev. E 73, 025101(R) (2006)\nKlemm, K., Stadler, P.F.: A note on fundamental, non-fundamental, and robust cycle bases. Discrete Appl. Math. 157, 2432–2438 (2009)\nLin, Y.J.: Connected sum construction of constant \\(Q\\)-curvature maniolds in higher dimensions. Differ. Geom. Appl. 40, 290–320 (2015)\nLoebl, M., Matamala, M.: Some remarks on cycles in graphs and digraphs. Discret. Math. 233(1–3), 175–182 (2001)\nMac Lane, S.: Categories for the Working Mathematician, 2nd edn., Graduate Texts in Mathematics (Book 5). Springer, New York (1998)\nMilnor, J.: A unique decomposition theorem for 3-manifolds. Am. J. Math. 84, 1–7 (1962)\nOEIS Foundation Inc.: The On-Line encyclopedia of integer sequences. http:\u002F\u002Foeis.org\u002FA085408 (2018)\nOstermeier, P.-J., Hellmuth, M., Klemm, K., Leydold, J., Stadler, P.F.: A note on quasi-robust cycle bases. Ars Math. Contemp. 2, 231–240 (2009)\nSpanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1966)\nWall, C.T.C.: Classification problems in differential topology, V. Invent. Math. 1, 355–374 (1966) [corrigendum, ibid 2, 306 (1966)]\nWhite, A.T.: Graphs of Groups on Surfaces. Elsevier, Amsterdam (2001)\nWhitney, H.: Non-separable and planar graphs. Trans. AMS 34, 339–362 (1932)",{"EN":320},"Given two cycles A and B in a graph, such that \n                  \n                    \n                  \n                  $$A\\cap B$$\n                  \n                    \n                  \n                 is a non-trivial path, the connected sum \n                  \n                    \n                  \n                  $$A\\hat{+} B$$\n                  \n                    \n                  \n                 is the cycle whose edges are the symmetric difference of E(A) and E(B). A special kind of cycle basis for a graph, a connected sum basis, is defined. Such a basis has the property that a hierarchical method, building successive cycles through connected sum, eventually reaches all the cycles of the graph. It is proved that every graph has a connected sum basis. A property is said to be cooperative if it holds for the connected sum of two cycles when it holds for the summands. Cooperative properties that hold for the cycles of a connected sum basis will hold for all cycles in the graph. As an application, commutativity of a groupoid diagram follows from commutativity of a connected sum basis for the underlying graph of the diagram. An example is given of a noncommutative diagram with a (non-connected sum) basis of cycles which do commute.",{"EN":322},"Graph Bases and Diagram Commutativity",{"VOID":324},"10.1007\u002Fs00373-018-1891-y","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00373-018-1891-y",[327,342],{"id":328,"sortIndex":21,"researcher":20,"roles":329,"affiliations":330,"properties":339},"81c3ecd8-fcc7-4f4a-8c49-bbc398b7f612",[240],[331],{"id":20,"sortIndex":21,"affiliation":332,"properties":20},{"id":333,"createTime":334,"updateTime":334,"relativeEntities":335,"slug":20,"properties":336,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"7ae57769-f62f-47eb-96de-c5cc90b5c5b7","2024-01-20T12:33:24.467+00:00",[],{"title":337},{"VI":338},"Department of Mathematics, Box 2014, Virginia Commonwealth University, Richmond, USA",{"title":340},{"VI":341},"Richard H. Hammack",{"id":343,"sortIndex":256,"researcher":20,"roles":344,"affiliations":345,"properties":354},"94f76c96-23d0-4178-9d87-feb7a6ae377a",[240],[346],{"id":20,"sortIndex":21,"affiliation":347,"properties":20},{"id":348,"createTime":349,"updateTime":349,"relativeEntities":350,"slug":20,"properties":351,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"97b37a87-e060-4734-9a4f-7589d8f5fe2e","2023-12-18T11:34:56.834+00:00",[],{"title":352},{"VI":353},"Department of Mathematics and Statistics, Georgetown University, Washington, USA",{"title":355},{"VI":356},"Paul C. 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American Elsevier, New York (1976)\nCatlin, P.A.: A reduction method to find spanning eulerian subgraphs. J. Graph Theory 12, 29–44 (1988)\nCatlin, P.A.: Supereulerian graphs, collapsible graphs, and four-cycles. Congr. Num. 58, 233–246 (1987)\nCatlin, P.A.: Double cycle covers and the Petersen graph. J. Graph Theory 13, 465–483 (1989)\nCatlin, P.A.: Supereulerian graphs: a survey. J. Graph Theory 16, 177–196 (1992)\nCatlin, P.A.: Double cycle covers and the Petersen graph. II. Congr. Num. 76, 173–181 (1990)\nCatlin, P.A., Lai, H.-J.: Supereulerian graphs and the Petersen graph. J. Combin. Theory, Ser B 66, 123–139 (1996)\nCavicchioli, A., Murgolo, T.E., Ruini, B., Spaggiari, F.: Special classes of snarks. Acta Appl. Math. 76, 57–88 (2003)\nCollier, J.B., Schmeichel, E.F.: New flip-flop constructions for hypohamiltonian graphs. Discret. Math. 18, 265–271 (1977)\nHolton, D.A., Sheehan, J.: The Petersen graph. Cambridge University Press, Cambridge (1993)\nJaeger, F.: A note on sub-Eulerian graphs. J. Graph Theory 3, 91–93 (1979)\nLi, X., Lei, L., Lai, H.-J., Zhang, M.: Supereulerian graphs and the Petersen graphs. Acta Math. Sin. English Ser. 30(2), 291–304 (2014)\nMáčajová, E., Škoviera, M.: Infinitely many hypohamiltonian cubic graphs of girth 7. Graphs Comb. 27, 231–241 (2011)\nYang, W.H.: Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013. http:\u002F\u002Ftel.archives-ouvertes.fr\u002Fdocs\u002F00\u002F87\u002F77\u002F93\u002FPDF\u002FVD2_YANG_WEIHUA_27092013",{"EN":401},"A graph G is hypohamiltonian if it is not Hamiltonian but for each \n                  \n                    \n                  \n                  $$v\\in V(G)$$\n                  \n                    \n                  \n                , the graph \n                  \n                    \n                  \n                  $$G-v$$\n                  \n                    \n                  \n                 is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset \n                  \n                    \n                  \n                  $$R\\subseteq V(G)$$\n                  \n                    \n                  \n                , there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173–181, 1990] and in [J. Combin. Theory, Ser B 66:123–139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291–304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.",{"EN":403},"Snarks, Hypohamiltonian Graphs and Non-Supereulerian Graphs",{"VOID":405},"10.1007\u002Fs00373-016-1718-7","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs00373-016-1718-7",[408],{"id":409,"sortIndex":21,"researcher":20,"roles":410,"affiliations":411,"properties":420},"bb3735b8-06af-466f-b65f-3c4e4cc9d0de",[240],[412],{"id":20,"sortIndex":21,"affiliation":413,"properties":20},{"id":414,"createTime":415,"updateTime":415,"relativeEntities":416,"slug":20,"properties":417,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"c00088bf-fb6d-4c38-8309-43095ba77209","2024-01-20T02:57:51.882+00:00",[],{"title":418},{"VI":419},"Butler University, Indianapolis, USA",{"title":421},{"VI":422},"Zhi-Hong Chen",{"url":406,"publisher":424,"properties":452},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":425,"slug":10,"properties":426,"entityType":18,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"subjectFields":430,"manageAffiliations":431,"indexDatabases":432,"url":20,"thumbnailPath":20,"statistic":447,"gsStatistic":20,"type":20,"analyzePriority":20},[],{"issn":427,"eissn":428,"title":429},{"VOID":13},{"VOID":15},{"EN":17},[],[],[433,440],{"id":66,"indexDatabase":434,"url":79,"indexYears":80,"academicFieldIds":439,"indexDatabaseRanking":84},{"id":68,"createTime":69,"updateTime":70,"relativeEntities":435,"label":436,"description":437,"key":76,"publicationTags":438,"standard":20},[],{"EN":73,"VI":73},{"EN":73,"VI":75},[78],[82,83],{"id":86,"indexDatabase":441,"url":101,"indexYears":20,"academicFieldIds":446,"indexDatabaseRanking":20},{"id":88,"createTime":89,"updateTime":90,"relativeEntities":442,"label":443,"description":444,"key":97,"publicationTags":445,"standard":20},[],{"EN":93,"VI":93},{"VI":95,"EN":96},[99,100],[103],{"impactFactor":21,"impactFactorByYear":448,"i10Index":114,"i10IndexLast5Year":115,"totalPublication":116,"totalPublicationByYear":449,"totalCitation":152,"totalCitationByYear":450,"totalCitationPerPublication":176,"totalCitationPerPublicationByYear":451,"hindexLast5Year":120,"hindex":120},{"2012":106,"2013":107,"2014":108,"2015":109,"2016":110,"2017":107,"2018":111,"2019":111,"2020":112,"2021":113,"2022":111,"2023":106},{"1985":118,"1986":119,"1987":120,"1988":121,"1989":122,"1990":123,"1991":124,"1992":125,"1993":126,"1994":127,"1995":124,"1996":128,"1997":129,"1998":124,"1999":130,"2000":120,"2001":131,"2002":132,"2003":133,"2004":127,"2005":134,"2006":130,"2007":135,"2008":136,"2009":137,"2010":138,"2011":139,"2012":140,"2013":141,"2014":142,"2015":143,"2016":144,"2017":145,"2018":146,"2019":147,"2020":148,"2021":149,"2022":150,"2023":151,"2024":127},{"1985":142,"1986":154,"1987":155,"1988":156,"1989":157,"1990":158,"1991":159,"1992":160,"1993":115,"1994":161,"1995":162,"1996":163,"1997":126,"1998":164,"1999":119,"2000":146,"2001":133,"2002":150,"2003":165,"2004":166,"2005":146,"2006":167,"2007":168,"2008":130,"2009":169,"2010":170,"2011":147,"2012":171,"2013":172,"2014":173,"2015":174,"2016":139,"2017":135,"2018":175,"2019":158,"2020":123,"2021":118,"2022":136,"2023":115},{"1985":178,"1986":179,"1987":180,"1988":181,"1989":182,"1990":183,"1991":184,"1992":185,"1993":186,"1994":187,"1995":188,"1996":189,"1997":190,"1998":191,"1999":192,"2000":193,"2001":194,"2002":195,"2003":196,"2004":197,"2005":198,"2006":193,"2007":199,"2008":200,"2009":201,"2010":202,"2011":203,"2012":204,"2013":205,"2014":206,"2015":207,"2016":208,"2017":179,"2018":209,"2019":204,"2020":210,"2021":209,"2022":211,"2023":212},{"volume":453,"pages":455},{"VOID":454},"32",{"VOID":456},"2267-2273","2016-05-20",2016,{"id":460,"createTime":461,"updateTime":462,"relativeEntities":463,"slug":464,"properties":465,"entityType":232,"verifyStatus":233,"verifyTime":462,"verifyNote":234,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":474,"fullTextUrl":20,"authors":475,"publicationType":272,"publisherRelationship":507,"citationCount":20,"citationInfo":20,"publishDate":541,"publishYear":542,"citationAnalyzeStatus":19,"lastCitationAnalyze":20,"indexDatabases":20,"openAccess":20,"references":20,"isForceReanalyzing":309},"102367c7-d885-4cc9-a350-7f8212f8b780","2024-02-09T20:01:14.105+00:00","2025-01-27T23:57:38.508+00:00",[],"Codings-of-graphs-with-binary-edge-labels",{"references":466,"abstract":468,"title":470,"doi":472},{"VOID":467},"Aigner, M., Triesch, E.: Irregular assignments of trees and forests. SIAM J. Discret Math.3, 439–449 (1990)\nAigner, M., Triesch, E.: Irregular assignments and two problems à la Ringel. In: Topics in combinatorics and graph theory (R. Bodendieck, R. Henn, eds.). Physics Verlag 1990, 29–36\nGolomb, S. W.: How to number a graph. In: Graph theory and computing (R.C. Read, ed.). Academic Press 1972, 23–37\nHedge, S.M.: Set-sequential graphs. to be published\nLehel, J.: Facts and quests on degree irregular assignments. to be published\nRingel, G.: Problem No. 25. In: Theory of graphs and its applications. Proc. Symp. Smolenice 1963 (M. Fiedler, ed.). Publ. House Czech. Acad. Sciences 1964\nTuza, Z.: Encoding the vertices of a graph with binary edge-labels. In: Sequences-combinatorics, compression, security and transmission (R.M. Capocelli, ed.). Springer-Verlag 1990, 287–299",{"EN":469},"LetG(V,E) be a graph. A mappingf:E→{0,1}\n                  m\n                 is called a (binary) coding ofG, if the induced mapping\n                  \n                    \n                  \n                  \n$$g:V \\to \\{ 0,1\\} ^m ,g(\\upsilon ) = \\sum\\limits_{e \\mathrel\\backepsilon  v} {f(e)} $$\n\n                , assigns different vectors to the vertices. For the Boolean sum,f is called aB-code, and for the mod 2 sum anM-code. Letm\n\n                  B\n                \n(G) resp.m\n\n                  M\n                \n(G) be the smallest lengthm for whichB-codes resp.M-codes are possible. Trivially,m\n\n                  B\n                \n(G),m\n\n                  M\n                \n(G) ≥ ⌈log2|V|⌉. Improving results of Z. Tuza we showm\n\n                  B\n                (G)≤⌈log2|V|⌉ + 1,m\n\n                  M\n                \n(G)≤⌈log2|V|⌉+4.",{"EN":471},"Codings of graphs with binary edge labels",{"VOID":473},"10.1007\u002FBF01202464","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01202464",[476,491],{"id":477,"sortIndex":21,"researcher":20,"roles":478,"affiliations":479,"properties":488},"a3700ad7-88be-49cb-89aa-0509b82cdae9",[240],[480],{"id":20,"sortIndex":21,"affiliation":481,"properties":20},{"id":482,"createTime":483,"updateTime":483,"relativeEntities":484,"slug":20,"properties":485,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"5318ee95-c872-4404-854b-343a7b7f2a59","2024-01-09T11:26:38.221+00:00",[],{"title":486},{"VI":487},"II. Mathematisches Institut, Freie Universität Berlin, Berlin, Germany",{"title":489},{"VI":490},"Martin Aigner",{"id":492,"sortIndex":256,"researcher":20,"roles":493,"affiliations":494,"properties":504},"5093d130-cbd9-4c72-aefb-7d9199c4ba93",[240],[495],{"id":20,"sortIndex":21,"affiliation":496,"properties":20},{"id":497,"createTime":498,"updateTime":498,"relativeEntities":499,"slug":500,"properties":501,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"5126c411-e6e7-4b79-8b97-cb40aeac8318","2024-04-17T20:05:22.442+00:00",[],"Institut-f%C3%BCr-Diskrete-Mathematik-Universit%C3%A4t-Bonn-Bonn-Germany",{"title":502},{"EN":503},"Institut für Diskrete Mathematik, Universität Bonn, Bonn, Germany",{"title":505},{"VI":506},"Eberhard Triesch",{"url":474,"publisher":508,"properties":536},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":509,"slug":10,"properties":510,"entityType":18,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"subjectFields":514,"manageAffiliations":515,"indexDatabases":516,"url":20,"thumbnailPath":20,"statistic":531,"gsStatistic":20,"type":20,"analyzePriority":20},[],{"issn":511,"eissn":512,"title":513},{"VOID":13},{"VOID":15},{"EN":17},[],[],[517,524],{"id":66,"indexDatabase":518,"url":79,"indexYears":80,"academicFieldIds":523,"indexDatabaseRanking":84},{"id":68,"createTime":69,"updateTime":70,"relativeEntities":519,"label":520,"description":521,"key":76,"publicationTags":522,"standard":20},[],{"EN":73,"VI":73},{"EN":73,"VI":75},[78],[82,83],{"id":86,"indexDatabase":525,"url":101,"indexYears":20,"academicFieldIds":530,"indexDatabaseRanking":20},{"id":88,"createTime":89,"updateTime":90,"relativeEntities":526,"label":527,"description":528,"key":97,"publicationTags":529,"standard":20},[],{"EN":93,"VI":93},{"VI":95,"EN":96},[99,100],[103],{"impactFactor":21,"impactFactorByYear":532,"i10Index":114,"i10IndexLast5Year":115,"totalPublication":116,"totalPublicationByYear":533,"totalCitation":152,"totalCitationByYear":534,"totalCitationPerPublication":176,"totalCitationPerPublicationByYear":535,"hindexLast5Year":120,"hindex":120},{"2012":106,"2013":107,"2014":108,"2015":109,"2016":110,"2017":107,"2018":111,"2019":111,"2020":112,"2021":113,"2022":111,"2023":106},{"1985":118,"1986":119,"1987":120,"1988":121,"1989":122,"1990":123,"1991":124,"1992":125,"1993":126,"1994":127,"1995":124,"1996":128,"1997":129,"1998":124,"1999":130,"2000":120,"2001":131,"2002":132,"2003":133,"2004":127,"2005":134,"2006":130,"2007":135,"2008":136,"2009":137,"2010":138,"2011":139,"2012":140,"2013":141,"2014":142,"2015":143,"2016":144,"2017":145,"2018":146,"2019":147,"2020":148,"2021":149,"2022":150,"2023":151,"2024":127},{"1985":142,"1986":154,"1987":155,"1988":156,"1989":157,"1990":158,"1991":159,"1992":160,"1993":115,"1994":161,"1995":162,"1996":163,"1997":126,"1998":164,"1999":119,"2000":146,"2001":133,"2002":150,"2003":165,"2004":166,"2005":146,"2006":167,"2007":168,"2008":130,"2009":169,"2010":170,"2011":147,"2012":171,"2013":172,"2014":173,"2015":174,"2016":139,"2017":135,"2018":175,"2019":158,"2020":123,"2021":118,"2022":136,"2023":115},{"1985":178,"1986":179,"1987":180,"1988":181,"1989":182,"1990":183,"1991":184,"1992":185,"1993":186,"1994":187,"1995":188,"1996":189,"1997":190,"1998":191,"1999":192,"2000":193,"2001":194,"2002":195,"2003":196,"2004":197,"2005":198,"2006":193,"2007":199,"2008":200,"2009":201,"2010":202,"2011":203,"2012":204,"2013":205,"2014":206,"2015":207,"2016":208,"2017":179,"2018":209,"2019":204,"2020":210,"2021":209,"2022":211,"2023":212},{"volume":537,"pages":539},{"VOID":538},"10",{"VOID":540},"1-10","1994-03-01",1994,{"id":544,"createTime":545,"updateTime":546,"relativeEntities":547,"slug":548,"properties":549,"entityType":232,"verifyStatus":233,"verifyTime":546,"verifyNote":234,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":558,"fullTextUrl":20,"authors":559,"publicationType":272,"publisherRelationship":587,"citationCount":20,"citationInfo":20,"publishDate":620,"publishYear":308,"citationAnalyzeStatus":19,"lastCitationAnalyze":20,"indexDatabases":20,"openAccess":20,"references":20,"isForceReanalyzing":309},"c27bc28b-1c38-449e-8f74-79b1a85bd153","2024-01-12T07:08:12.140+00:00","2024-12-25T23:56:50.781+00:00",[],"-M-24-Orbits-of-Octad-Triples",{"references":550,"abstract":552,"title":554,"doi":556},{"VOID":551},"Aschbacher, M.: Sporadic Groups. Cambridge Tracts in Mathematics, vol. 104. Cambridge University Press, Cambridge (1994)\nBoston, N.: Private Communication\nBoston, N.: A multivariate weight enumerator for tail-biting trellis pseudocodewords. J. Gen. Lie Theory Appl. 9(S1) (2015) (Art. ID S1-004)\nBrouwer, A.E., Cuypers, H., Lambeck, E.W.: The hyperplanes of the \\(M_{24}\\) near polygon. Graphs Comb. 18(3), 415–420 (2002)\nCalderbank, A.R., Forney Jr., G.D., Vardy, A.: Minimal tail-biting trellises: the Golay code and more. IEEE Trans. Inf. Theory 45(5), 1435–1455 (1999)\nChoi, C.: On subgroups of \\(M_{24}\\). I. Stabilizers of subsets. Trans. Am. Math. Soc. 167, 1–27 (1972)\nConway, J.H.: Three lectures on exceptional groups. In: Finite Simple Groups (Proceedings of Instructional Conference, Oxford, 1969), pp. 215–247. Academic Press, London (1971)\nConway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. With Additional Contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290. Springer, New York (1999)\nCurtis, R.T.: A new combinatorial approach to \\(M_{24}\\). Math. Proc. Camb. Philos. Soc. 79(1), 25–42 (1976)\nCurtis, R.T.: The maximal subgroups of \\(M_{24}\\). Math. Proc. Camb. Philos. Soc. 81(2), 185–192 (1977)\nGolay, M.J.E.: Notes on digital coding. Proc. IRE 37, 657 (1949)\nRonan, M.A., Smith, S.D.: 2-local geometries for some sporadic groups. In: The Santa Cruz Conference on Finite Groups (University of California, Santa Cruz, California, 1979), Proceedings of Symposia in Pure Mathematics, vol. 37, pp. 283–289. American Mathematical Society, Providence (1980)\nWitt, E.: Die 5-fach transitiven gruppen von mathieu. Abh. Math. Sem. Univ. Hambg. 12(1), 256–264 (1937)\nWitt, E.: Über Steinersche Systeme. Abh. Math. Sem. Univ. Hambg. 12(1), 265–275 (1937)",{"EN":553},"An octad triple is a set of three octads, octads being the blocks of the S(5, 8, 24) Steiner system. In this paper we determine the orbits of \n                  \n                    \n                  \n                  $$M_{24}$$\n                  \n                    \n                  \n                , the largest Mathieu group, upon the set of octad triples.",{"EN":555},"$$M_{24}$$ -Orbits of Octad 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I., Álvarez-Rebollar, J.L., Catana-Salazar, J.C., Jiménez-Salinas, M., Solís-Villarreal, E., Urrutia, J.: Minimizing the solid angle sum of orthogonal polyhedra and guarding them with \\(\\frac{\\pi }{2}\\)-edge guards. In: Proceedings of the 28th Canadian Conference on Computational Geometry, Vancouver, August 3-5, pp. 175–181 (2016)\nAldana-Galván, I., Álvarez-Rebollar, J.L., Catana-Salazar, J.C., Marín-Nevárez, N., Solís-Villarreal, E., Urrutia, J., Velarde, C.: Covering orthotrees with guards and beacons. In: In proceedings of XVII Spanish Meeting on Computational Geometry, Alicante, Spain, July 26–28, pp. 56–68 (2017)\nBae, S.W., Shin, C.S., Vigneron, A.E.: Tight bounds for beacon-based coverage in simple rectilinear polygons. To appear in Computational Geometry (2019). https:\u002F\u002Fdoi.org\u002F10.1016\u002Fj.comgeo.2019.02.002\nBenbernou, N.M., Demaine, E.D., Demaine, M.L., Kurdia, A., O’Rourke, J., Toussaint, G., Urrutia, J., Viglietta, G.: Edge-guarding orthogonal polyhedra. In: Proceedings of the 23rd Canadian Conference on Computational Geometry, Toronto, August 10–12, pp. 461–466 (2011)\nBiro, M.: Beacon-based routing and guarding. Ph.D. thesis, State University of New York at Stony Brook (2013)\nBiro, M., Gao, J., Iwerks, J.B., Kostitsyna, I., Mitchell, J.S.: Beacon-based routing and coverage. In: 21st Fall Workshop on Computational Geometry, New York, November 4–5 (2011)\nChvátal, V.: A combinatorial theorem in plane geometry. J. Comb Theory Ser B 18(1), 39–41 (1975)\nCleve, J.: Combinatorics of beacon-based routing and guarding in three dimensions. Master’s thesis, Freie Universität Berlin (2017)\nCleve, J., Mulzer, W.: Combinatorics of beacon-based routing in three dimensions. In: Latin American Symposium on Theoretical Informatics, Buenos Aires, April 16-19, pp. 346–360. Springer (2018)\nDamian, M., Flatland, R.: Unfolding low-degree orthotrees with constant refinement. In: 30th Canadian Conference on Computational Geometry, Winnipeg, August 8–10. Elsevier (2018)\nDamian, M., Flatland, R.: Unfolding orthotrees with constant refinement. arXiv preprint arXiv:1811.01842 (2018)\nIwerks, J.G.: Combinatorics and complexity in geometric visibility problems. Ph.D. thesis, State University of New York at Stony Brook (2012)\nO’Rourke, J.: Art Gallery Theorems and Algorithms, vol. 57. Oxford University Press Inc., New York (1987)\nShermer, T.C.: Recent results in art galleries. Proc. IEEE 80, 1384–1384 (1992)\nShermer, T.C.: A combinatorial bound for beacon-based routing in orthogonal polygons. (2015) arXiv:1507.03509\nShermer, T.C.: A combinatorial bound for beacon-based routing in orthogonal polygons. In: Proceedings of the 27th Canadian Conference on Computational Geometry, Kingston, Ontario, August 10–12, pp. 213–219 (2015)\nTomás, A.P.: Guarding thin orthogonal polygons is hard. In: International Symposium on Fundamentals of Computation Theory, pp. 305–316. Springer (2013)\nUrrutia, J.: Art gallery and illumination problems. In: Handbook of computational geometry, pp. 973–1027. Elsevier (2000)\nViglietta, G.: Guarding and searching polyhedra. Ph.D. thesis, University of Pisa (2012)\nViglietta, G.: Face-guarding polyhedra. Comput. Geom. 47(8), 833–846 (2014)",{"EN":631},"We consider two variants of the Art Gallery Problem: illuminating orthotrees with a minimum set of vertex lights, and covering orthotrees with a minimum set of vertex beacons. An orthotree P is a simply connected orthogonal polyhedron that is the union of a set S of cuboids glued face to face such that the graph whose vertices are the cuboids of S, two of which are adjacent if they share a common face, is a tree. A point p illuminates a point $$q \\in P$$ if the line segment $$\\ell$$ joining them is contained in P. A beacon b is a point in P that pulls other points in P towards itself similarly to the way a magnet attracts ferrous particles. We say that a beacon bcoversp if when b starts pulling p, p does not get stuck at a point of P before it reaches b. This happens, for instance if p reaches a point $$p'$$ such that there is an $$\\epsilon >0$$ such that any point in P at distance at most $$\\epsilon$$ from $$p'$$ is farther away from $$p'$$ than q (there is another pathological case that we will not detail in this abstract). In this paper we prove that any orthotree P with n vertices can be illuminated using at most $$\\lfloor n\u002F8 \\rfloor$$ light sources placed at vertices of P, and that all of the points in P can always be covered with at most $$\\lfloor n\u002F12 \\rfloor$$ vertex beacons. Both bounds are tight.",{"EN":633},"Tight Bounds for Illuminating and Covering of Orthotrees with Vertex Lights and Vertex Beacons",{"VOID":635},"10.1007\u002Fs00373-020-02141-4","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00373-020-02141-4",[638,653,665,677,690,705,723],{"id":639,"sortIndex":183,"researcher":20,"roles":640,"affiliations":641,"properties":650},"dedc7130-65c9-413e-bc64-b3418bb6fc6e",[240],[642],{"id":20,"sortIndex":21,"affiliation":643,"properties":20},{"id":644,"createTime":645,"updateTime":645,"relativeEntities":646,"slug":20,"properties":647,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"67531b9e-b4f6-4df8-bf38-d247689e0078","2024-01-10T11:55:42.109+00:00",[],{"title":648},{"VI":649},"Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México, Mexico City, México",{"title":651},{"VI":652},"J. C. 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Álvarez-Rebollar",{"url":636,"publisher":739,"properties":767},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":740,"slug":10,"properties":741,"entityType":18,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"subjectFields":745,"manageAffiliations":746,"indexDatabases":747,"url":20,"thumbnailPath":20,"statistic":762,"gsStatistic":20,"type":20,"analyzePriority":20},[],{"issn":742,"eissn":743,"title":744},{"VOID":13},{"VOID":15},{"EN":17},[],[],[748,755],{"id":66,"indexDatabase":749,"url":79,"indexYears":80,"academicFieldIds":754,"indexDatabaseRanking":84},{"id":68,"createTime":69,"updateTime":70,"relativeEntities":750,"label":751,"description":752,"key":76,"publicationTags":753,"standard":20},[],{"EN":73,"VI":73},{"EN":73,"VI":75},[78],[82,83],{"id":86,"indexDatabase":756,"url":101,"indexYears":20,"academicFieldIds":761,"indexDatabaseRanking":20},{"id":88,"createTime":89,"updateTime":90,"relativeEntities":757,"label":758,"description":759,"key":97,"publicationTags":760,"standard":20},[],{"EN":93,"VI":93},{"VI":95,"EN":96},[99,100],[103],{"impactFactor":21,"impactFactorByYear":763,"i10Index":114,"i10IndexLast5Year":115,"totalPublication":116,"totalPublicationByYear":764,"totalCitation":152,"totalCitationByYear":765,"totalCitationPerPublication":176,"totalCitationPerPublicationByYear":766,"hindexLast5Year":120,"hindex":120},{"2012":106,"2013":107,"2014":108,"2015":109,"2016":110,"2017":107,"2018":111,"2019":111,"2020":112,"2021":113,"2022":111,"2023":106},{"1985":118,"1986":119,"1987":120,"1988":121,"1989":122,"1990":123,"1991":124,"1992":125,"1993":126,"1994":127,"1995":124,"1996":128,"1997":129,"1998":124,"1999":130,"2000":120,"2001":131,"2002":132,"2003":133,"2004":127,"2005":134,"2006":130,"2007":135,"2008":136,"2009":137,"2010":138,"2011":139,"2012":140,"2013":141,"2014":142,"2015":143,"2016":144,"2017":145,"2018":146,"2019":147,"2020":148,"2021":149,"2022":150,"2023":151,"2024":127},{"1985":142,"1986":154,"1987":155,"1988":156,"1989":157,"1990":158,"1991":159,"1992":160,"1993":115,"1994":161,"1995":162,"1996":163,"1997":126,"1998":164,"1999":119,"2000":146,"2001":133,"2002":150,"2003":165,"2004":166,"2005":146,"2006":167,"2007":168,"2008":130,"2009":169,"2010":170,"2011":147,"2012":171,"2013":172,"2014":173,"2015":174,"2016":139,"2017":135,"2018":175,"2019":158,"2020":123,"2021":118,"2022":136,"2023":115},{"1985":178,"1986":179,"1987":180,"1988":181,"1989":182,"1990":183,"1991":184,"1992":185,"1993":186,"1994":187,"1995":188,"1996":189,"1997":190,"1998":191,"1999":192,"2000":193,"2001":194,"2002":195,"2003":196,"2004":197,"2005":198,"2006":193,"2007":199,"2008":200,"2009":201,"2010":202,"2011":203,"2012":204,"2013":205,"2014":206,"2015":207,"2016":208,"2017":179,"2018":209,"2019":204,"2020":210,"2021":209,"2022":211,"2023":212},{"volume":768,"pages":770},{"VOID":769},"36",{"VOID":771},"617-630","2020-02-20",2020,{"id":775,"createTime":776,"updateTime":777,"relativeEntities":778,"slug":779,"properties":780,"entityType":232,"verifyStatus":233,"verifyTime":777,"verifyNote":234,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":789,"fullTextUrl":20,"authors":790,"publicationType":272,"publisherRelationship":847,"citationCount":20,"citationInfo":20,"publishDate":881,"publishYear":882,"citationAnalyzeStatus":19,"lastCitationAnalyze":20,"indexDatabases":20,"openAccess":20,"references":20,"isForceReanalyzing":309},"934007be-96a9-4950-87bb-36bbcd0c0311","2024-01-22T15:31:18.306+00:00","2024-12-12T23:54:13.881+00:00",[],"Small-Edge-Sets-Meeting-all-Triangles-of-a-Graph",{"references":781,"abstract":783,"title":785,"doi":787},{"VOID":782},"Aharoni R.: Ryser’s conjecture for tripartite 3-graphs. Combinatorica 21, 1–4 (2001)\nLakshmanan, S. Aparna, Bujtás, Cs., Tuza, Zs.: manuscript in preparation\nBagga J.: Old and new generalizations of line graphs. Int. J. Math. Math. Sci 29, 1509–1521 (2004)\nChapuy, G., DeVos, M., McDonald, J., Mohar, B., Scheide, D.: Packing triangles in weighted graphs (2010)\nChudnovsky M., Robertson N., Seymour P., Thomas R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)\nCui Q., Haxell P., Ma W.: Packing and covering triangles in planar graphs. Graphs Combin. 25, 817–824 (2009)\nErdős P., Gallai T., Tuza Zs.: Covering and independence in triangle structures. Discret. Math. 150, 89–101 (1996)\nHaxell P.E.: Packing and covering triangles in graphs. Discret. Math. 195, 251–254 (1999)\nHaxell P.E., Kohayakawa Y.: Packing and covering triangles in tripartite graphs. Graphs Combin. 14, 1–10 (1998)\nHaxell, P., Kostochka, A., Thomassé, S.: A stability theorem on fractional covering of triangles by edges (2010)\nHolyer I.: The NP-completeness of edge-colouring. SIAM J. Comput. 10, 718–720 (1981)\nKrivelevich M.: On a conjecture of Tuza about packing and covering of triangles. Discret. Math. 142, 281–286 (1995)\nLe V.B.: Gallai graphs and anti-Gallai graphs. Discret. Math. 159, 179–189 (1996)\nMansour T., Song C., Yuster R.: A comment on Ryser’s conjecture for intersecting hypergraphs. Graphs Combin. 25, 101–109 (2009)\nPrisner E.: Intersection multigraphs of uniform hypergraphs. Graphs Combin. 14, 363–375 (1998)\nTuza, Zs.: Conjecture, finite and infinite sets. In: Hajnal, A., Lovász, L., Sós, V.T. (eds.) Proc. Colloq. Math. Soc. J. Bolyai (Eger, Hungary, 1981), vol. 37, p. 888, North-Holland, Amsterdam (1984)\nTuza Zs.: Ryser’s conjecture on transversals of r-partite hypergraphs. Ars Combin. 16B, 201–209 (1983)\nTuza Zs.: A conjecture on triangles of graphs. Graphs Combin. 6, 373–380 (1990)\nTuza, Zs.: Some open problems on colorings and coverings of graphs (Abstract), Graphentheorie-Tagung Oberwolfach (1990)\nTuza Zs.: Perfect triangle families. Bull. Lond. Math. Soc. 26, 321–324 (1994)",{"EN":784},"It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for ‘triangle-3-colorable’ graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K\n                4-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.",{"EN":786},"Small Edge Sets Meeting all Triangles of a Graph",{"VOID":788},"10.1007\u002Fs00373-011-1048-8","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00373-011-1048-8",[791,806,823],{"id":792,"sortIndex":21,"researcher":20,"roles":793,"affiliations":794,"properties":803},"4b0ef243-14eb-453c-aebc-37126d82c8de",[240],[795],{"id":20,"sortIndex":21,"affiliation":796,"properties":20},{"id":797,"createTime":798,"updateTime":798,"relativeEntities":799,"slug":20,"properties":800,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"dde5c8eb-687e-48f5-9857-2ed4bf52754a","2024-01-22T15:31:18.323+00:00",[],{"title":801},{"VI":802},"Department of Mathematics, St. Xavier’s College for Women, Aluva, India",{"title":804},{"VI":805},"S. 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Cambridge University Press, New York (1992)\nBrooks R.L.: On colouring the nodes of a network. Proc. Camb. Phil. Soc. 37, 194–197 (1941)\nDanziger P., Graham A., Mendelsohn E.: The chromatic spectrum of graph designs. Bull. Inst. Combin. Appl. 50, 71–96 (2007)\nDembowski P.: Finite Geometries, pp. 4–5. Springer-Verlag, New York (1968)\nGropp H.: Configurations. In: Colbourn, C.J., Dinitz, J.H. (eds) The CRC Handbook of Combinatorial Designs., pp. 253–255. CRC Press, Boca Raton (1996)\nHarary F., Robinson R.W., Wormald N.C.: Isomorphic factorisations I. Complete graphs. Trans. Am. Math. Soc. 242, 243–260 (1978)\nJamison R.E., Mendelsohn E.: On the chromatic spectrum of acyclic decompositions of graphs. J. Graph Theory 56, 83–104 (2007)\nJamison R.E., White J.H.: On intersection graphs of 2-matchings in cubic graphs. Congr. Numerantium 181, 187–193 (2006)\nWest D.B.: Introduction to Graph Theory, 2nd edn. 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J. Graph Theory 57(3), 255–264 (2007)\nBarish, R.D., Suyama, A.: Counting Hamiltonian cycles on quartic 4-vertex-connected planar graphs. In: Online collection of abstracts for the 20th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG3), pp. 129–130. http:\u002F\u002Fwww.jcdcgg.u-tokai.ac.jp\u002F (2017). Accessed 30 Sept 2017\nBarish, R. D.: Personal communication (2017)\nBondy, J.A., Jackson, B.: Vertices of small degree in uniquely Hamiltonian graphs. J. Combin. Theory Ser. B 74(2), 265–275 (1998)\nBriquel, I., Koiran, P.: A dichotomy theorem for polynomial evaluation. In: Proceedings of the 34th annual symposium on Mathematical Foundations of Computer Science (MFCS), pp. 187–198 (2009)\nChiba, N., Nishizeki, T.: The Hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. J. Algorithms 10(2), 187–211 (1989)\nDyer, M., Goldberg, L.A., Greenhill, C., Jerrum, M.: The relative complexity of approximate counting problems. Algorithmica 38(3), 471–500 (2004)\nFleischner, H.: Uniquely Hamiltonian graphs of minimum degree 4. J. Graph Theory 75(2), 167–177 (2014)\nGarey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)\nGarey, M.R., Johnson, D.S., Tarjan, R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4), 704–714 (1976)\nKarp, R. M., Luby, M.: Monte-Carlo algorithms for enumeration and reliability problems. In: Proceedings of the 24th Annual Symposium on Foundations of Computer Science (FOCS), pp. 56–64 (1983)\nLiśkiewicz, M., Ogihara, M., Toda, S.: The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theor. Comput. Sci. 304(1–3), 129–156 (2003)\nMenger, K.: Zur allgemeinen kurventheorie. Fundamenta Mathematicae 10(1), 96–115 (1927)\nMeredith, G.H.J.: Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs. J. Combin. Theory Ser. B 14(1), 55–60 (1973)\nPetersen, J.: Die theorie der regulären graphs. Acta Math. 15, 193–220 (1891)\nSheehan, J.: The multiplicity of Hamiltonian circuits in a graph. In: Recent Advances in Graph Theory. Proceedings of the 2nd Czechoslovak Symposium, Prague, 1974, pp. 477–480 (1975)\nSteinitz, E.: Polyeder und raumeinteilungen. Encyklopädie der mathematischen Wissenschaften 3(part 1.2, B)), 1–139 (1922)\nTait, P.G.: Listing’s topologie. Philosophical Magazine (5th ser.) 17, 30–46 (1884)\nThomassen, C.: A theorem on paths in planar graphs. J. Graph Theory 7, 169–176 (1983)\nTutte, W.T.: On Hamiltonian circuits. J. Lond. Math. Soc. 21, 98–101 (1946)\nTutte, W.T.: A theorem on planar graphs. Trans. Am. Math. Soc. 82, 99–116 (1956)\nValiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979)\nValiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)\nWhitney, H.: A theorem on graphs. Ann. Math. 32(2), 378–390 (1931)\nZanko, V.: \\(\\#P\\)-completeness via many-one reductions. Int. J. Found. Comput. Sci. 2(1), 77–82 (1991)\nZuckerman, D.: On unapproximable versions of NP-complete problems. SIAM J. 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