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

Edge-dominating cycles in graphs

AbstractA set S of vertices in a graph G is said to be an edge-dominating set if every edge in G is incident with a vertex in S. A cycle in G is said to be a dominating cycle if its vertex set is an edge-dominating set. Nash-Williams [Edge-disjoint hamiltonian circuits in graphs with vertices of large valency, Studies in Pure Mathematics, Academic Press, London, 1971, pp. 157–183] has proved that every longest cycle in a 2-connected graph of order n and minimum degree at least 13(n+2) is a dominating cycle. In this paper, we prove that for a prescribed positive integer k, under the same minimum degree condition, if n is sufficiently large and if we take k disjoint cycles so that they contain as many vertices as possible, then these cycles form an edge-dominating set. Nash-Williams’ Theorem corresponds to the case of k=1 of this result

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