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arXiv · 2610.33041

The minimum number of wheel-saturating edges

Xianglei Chen·Changxiang He
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Abstract

Let $G$ be an $H$-free graph and $e$ be a non-edge of $G$. We say $e$ is an $H$-saturating edge of $G$, if $G+e$ contains a copy of $H$. We denote by $f_H(n,m)$ the minimum number of $H$-saturating edges of an $n$-vertex $H$-free graph with exactly $m$ edges. In this paper, we show that $f_{W_{2k+1}}(n,ex(n,F_{2k+1})+1)= \begin{cases} \ell -k, &(k-2)\ell \ \text{is even}, 2(\ell-k)-1, &(k-2)\ell \ \text{is odd}, \end{cases}$ and $f_{W_{2k}}(n,ex(n,F_{2k})+1)=\left(\frac{2}{33}+o(1)\right)n^2.$ At the same time, we characterize the corresponding extremal graphs.

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Xianglei Chen, Changxiang He. 2026-10-16. The minimum number of wheel-saturating edges. https://arxiv.org/abs/2610.33041

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