The Ramsey Number for Tree Versus Wheel with Odd Order

TL;DR

This paper investigates the Ramsey number for trees versus wheels with odd order, providing new bounds and exact values for specific cases, especially when the tree's maximum degree is large.

Contribution

It determines the Ramsey number R(Tn,W8) for trees with maximum degree at least n-3 and refines the conjecture on small maximum degree cases.

Findings

R(Tn,W8) > 2n-1 for trees with maximum degree at least n-3

R(Tn,Wm) depends on both m and n when the maximum degree is large

Refined conditions for the conjecture on small maximum degree trees

Abstract

Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum degree of Tn is small and m is even. Related to the conjecture, it is interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In this paper, we find the Ramsey number R(Tn,W8) for tree Tn with the maximum degree of Tn is at least n-3, namely R(Tn,W8) > 2n-1 for almost all such tree Tn. We also prove that if the maximum degree of Tn is large, then R(Tn,Wm) is a function of both m and n. In the end, we refine the conjecture of Chen et al. by giving the condition of small maximum degree. For a tree Tn with large maximum degree and even m, the R(Tn,Wm) is also unknown in general. In this paper, we shall determine the Ramsey number R(Tn,W8) for all trees Tn of order n with the maximum degree of Tn is at least n-3.