Tóm tắt
An [n, k, r]-partite graph
is a graph whose vertex set, V, can be partitioned into n
pairwise-disjoint independent sets,
V1, …, Vn, each
containing exactly k vertices, and the subgraph induced by
Vi ∪ Vj
contains exactly r independent edges, for 1 [les ] i
< j [les ] n. An independent transversal
in an [n, k, r]-partite graph is
an independent set,
T, consisting of n vertices, one from each
Vi. An independent covering is a
set of k pairwise-disjoint independent transversals. Let
t(k, r) denote the maximal n for which
every
[n, k, r]-partite graph contains an
independent
transversal. Let c(k, r) be the maximal
n for which
every [n, k, r]-partite graph
contains an independent
covering. We give upper and lower bounds for these parameters.
Furthermore, our bounds are constructive. These results improve and
generalize previous results of Erdo″s, Gyárfás and
Łuczak [5], for the case of graphs.
