Coloring a graph with Δ1 colors: Conjectures equivalent to the Borodin–Kostochka conjecture that appear weaker

European Journal of Combinatorics - Tập 44 - Trang 23-42 - 2015
Daniel W. Cranston1, Landon Rabern2
1Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, United States
2School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States

Tài liệu tham khảo

Benedict, 1978, vol. 642, 132 O.V. Borodin, Criterion of chromaticity of a degree prescription, in: Abstracts of IV All-Union Conf. on Th. Cybernetics, 1977, pp. 127–128. Borodin, 1977, On an upper bound of a graph’s chromatic number, depending on the graph’s degree and density, J. Combin. Theory Ser. B, 23, 247, 10.1016/0095-8956(77)90037-5 Brooks, 1941, vol. 37, 194 Catlin, 1976 Catlin, 1979, Hajós graph-coloring conjecture: variations and counterexamples, J. Combin. Theory Ser. B, 26, 268, 10.1016/0095-8956(79)90062-5 D.W. Cranston, L. Rabern, Conjectures equivalent to the Borodin–Kostochka Conjecture that are a priori weaker, 2012. arXiv:1203.5380. D.W. Cranston, L. Rabern, Graphs with χ=Δ have big cliques, 2013. http://arxiv.org/abs/1305.3526. Emden-Weinert, 1998, Uniquely colourable graphs and the hardness of colouring graphs of large girth, Combin. Probab. Comput., 7, 375, 10.1017/S0963548398003678 Entringer, 1985, A short proof of Rubin’s block theorem, vol. 115, 367 P. Erdős, A.L. Rubin, H. Taylor, Choosability in graphs, in: Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol. 26, 1979, pp. 125–157. Hladkỳ, 2010, Brooks’ theorem via the Alon–Tarsi theorem, Discrete Math., 310, 3426, 10.1016/j.disc.2010.07.019 Jensen, 1995 Kierstead, 2000, On the choosability of complete multipartite graphs with part size three, Discrete Math., 211, 255, 10.1016/S0012-365X(99)00157-0 King, 2011, Hitting all maximum cliques with a stable set using lopsided independent transversals, J. Graph Theory, 67, 300, 10.1002/jgt.20532 King, 2012, A fractional analogue of Brooks’ theorem, SIAM J. Discrete Math., 26, 452, 10.1137/110827879 Kostochka, 1980, Degree, density, and chromatic number, Metody Diskret. Anal., 35, 45 Kostochka, 1996, The colour theorems of Brooks and Gallai extended, Discrete Math., 162, 299, 10.1016/0012-365X(95)00294-7 Lovász, 1966, On decomposition of graphs, SIAM J. Algebr. Discrete Methods, 3, 237 Molloy, 2002 Mozhan, 1983, Chromatic number of graphs with a density that does not exceed two-thirds of the maximal degree, Metody Diskretn. Anal., 39, 52 Rabern, 2011, On hitting all maximum cliques with an independent set, J. Graph Theory, 66, 32, 10.1002/jgt.20487 Reed, 1999, A strengthening of Brooks’ theorem, J. Combin. Theory Ser. B, 76, 136, 10.1006/jctb.1998.1891 Reed, 2004, List colouring when the chromatic number is close to the order of the graph, Combinatorica, 25, 117, 10.1007/s00493-005-0010-x