Year: 2026
Type: preprint
Abstract: For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for every possible chromatic number. Let $T_{n,r}$ be the complete $r$-partite Turán graph on $n$ vertices. We find a constant $γ_r>0$ such that if $n$ is sufficiently large and $\mathcal{H}$ is a finite family with $χ(\mathcal{H})=r+1$ satisfying \[ ex(n,\mathcal{H})
Source: arXiv (Cornell University)
Authors: Lin, Jifu, Wang, Xiaolin, Yu, Guangmiao
Institutions: East China Normal University, Fuzhou University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221173474