Balanced independent sets in graphs omitting large cliques

Abstract

Our goal is to investigate a close relative of the independent transversal problem in the class of infinite Kn-free graphs: we show that for any infinite Kn-free graph G " pV, Eq and m P N there is a minimal r " rpG, mq such that for any balanced r-colouring of the vertices of G one can find an independent set which meets at least m colour classes in a set of size |V |. Answering a conjecture of S. Thomassé, we express the exact value of rpHn, mq (using Ramsey-numbers for finite digraphs), where Hn is Henson's countable universal homogeneous Kn-free graph. In turn, we deduce a new partition property of Hn regarding balanced embeddings of bipartite graphs: for any finite bipartite G with bipartition A, B, if the vertices of Hn are partitioned into two infinite classes then there is an induced copy of G in Hn such that the images of A and B are contained in different classes.

Verification of Karci Algorithm’s Efficiency for Maximum Independent Set Problem in Graph Theory

“…Brandstadt and Mosca (Brandstadt and Mosca, 2018) used dynamic programming approach to show that the maximum weight independent set can be solved in polynomial time for claw-free graphs. Laflamme and his/her friends (Laflamme and et al, 2019) tried to show that Kn-free graph and minimal r=r(G,m) where mN, independent set meets at least m colour classes in a set of size |V| for any balanced r-colouring of the vertices of graph G. Lin et al obtained the number of independent sets and number of maximum independent set for path-tree bipartite graphs (Lin, 2018a), and Oh studied on the number of maximum independent sets for complete rectangular grid graph (Oh, 2017). Wan and his/her friends studied on independent sets and matchings of some special graphs (Wan et al, 2018).…”