Ramsey chains in graphs

Abstract

Let G be a graph with a red-blue coloring c of the edges of G . A Ramsey chain in G with respect to c is a sequence G 1 , G 2 , . . . , G k of pairwise edge-disjoint subgraphs of G such that each subgraph G i ( 1 ≤ i ≤ k ) is monochromatic of size i and G i is isomorphic to a subgraph of G i +1 ( 1 ≤ i ≤ k − 1 ). The Ramsey index AR c ( G ) of G with respect to c is the maximum length of a Ramsey chain in G with respect to c . The Ramsey index AR ( G ) of G is the minimum value of AR c ( G ) among all red-blue colorings c of G . A Ramsey chain with respect to c is maximal if it cannot be extended to one of greater length. The lower Ramsey index AR − c ( G ) of G with respect to c is the minimum length of a maximal Ramsey chain in G with respect to c . The lower Ramsey index AR − ( G ) of G is the minimum value of AR − c ( G ) among all red-blue colorings c of G . Ramsey chains and maximal Ramsey chains are investigated for stars, matchings, and cycles. It is shown that (1) for every two integers p and q with 2 ≤ p < q , there exists a graph with a red-blue coloring possessing a maximal Ramsey chain of length p and a maximum Ramsey chain of length q and (2) for every positive integer k , there exists a graph with a red-blue coloring possessing at least k maximal Ramsey chains of distinct lengths with prescribed conditions. A conjecture and additional results are also presented.