arXiv:2610.33041v1 Announce Type: new 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.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page