R. H. Schelp conjectured that if G is a graph with |V(G)| = R(P n, P n) such that δ(G) > 3|V(G)|/4, then in every 2-colouring of the edges of G there is a monochromatic P n. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(S n, nK 2, nK 2) = 3n - 1. This extends R(nK 2, nK 2) = 3n - 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma. It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.
ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics