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).
Categories: math.CO