Independent Transversals and Independent Coverings in Sparse Partite Graphs

An [n,k,r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V_l,...,V_n, each containing exactly k vertices, and the subgraph induced by V_i ∪ V_j 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 K_i. 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 [5], for the case of graphs.