An [n,k,r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, Vl,...,Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 ≤ i < j ≤ n. An independent transversal in an [n,k,r]-partite graph is an independent set, T, consisting of n vertices, one from each Ki. 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 Erdos, Gyárfás and Łuczak , for the case of graphs.
ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics