An exact Ramsey number of large bipartite graphs versus odd wheel

TL;DR

This paper determines the exact Ramsey number for large bipartite graphs ,n versus odd wheel graphs, revealing new structural insights and combining probabilistic and structural methods for the proof.

Contribution

It establishes the exact Ramsey number R(,n, W_m) = 3n+4 for large n and m, a previously unexplored case involving ,n and odd wheels.

Findings

Proved R(,n, W_m) = 3n+4 for large n,m with n m and m odd.

Resolved a structural rigidity question about highly dependent neighborhoods.

Combined probabilistic and structural analysis techniques.

Abstract

The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus the wheel is not yet explored due to the bit more structural complexity of $\mathbb{K}_{2,n}$ compared to the star. In this article, we have established an exact value of $\mathbb{K}_{2,n}$ versus $W_{m}$ for large $n$ and $m$. In particular, we have proved \begin{equation*} R(\mathbb{K}_{2,n}, W_{m})=3n+4, \end{equation*} whenever $n$ and $m$ are sufficiently large integers satisfying $n\geq4m$ and $m$ is an odd integer. This proves the $W_{m}$-goodness of $\mathbb{K}_{2,n}$. Our proof combines probabilistic methods with an analysis of structural dependencies. As part of the argument, we resolve a structural rigidity question concerning highly dependent neighbourhoods (Lemma 3.12).