Independence Number and Disjoint Theta Graphs

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs

Long Path Lemma Concerning Connectivity and Independence Number

We show that, in a k-connected graph G of order n with α(G)=α, between any pair of vertices, there exists a path P joining them with |P|≥min{n,(k−1)(n−k)/α +k}. This implies that, for any edge e∈E(G), there is a cycle containing e of length at least min{n,(k−1)(n−k)/α +k}. Moreover, we generalize our result as follows: for any choice S of s≤k vertices in G, there exists a tree T whose set of leaves is S with |T|≥min{n,(k−s+1)(n−k)/α +k}

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs