The circumference of a graph with given minimum degree and clique number

Abstract

The circumference denoted by $c(G)$ of a graph $G$ is the length of its longest cycle. Let $\delta(G)$ and $\omega(G)$ denote the minimum degree and the clique number of a graph $G$, respectively. In [\emph{Electron. J. Combin.} 31(4)(2024) $\#$P4.65], Yuan proved that if $G$ is a 2-connected graph of order $n$, then $c(G)\geq \min\{n,\omega(G)+\delta(G)\}$ unless $G$ is one of two specific graphs. In this paper, we prove a stability result for the theorem of Erd\H os and Gallai, thereby helping us to characterize all $2$-connected non-hamiltonian graphs whose circumference equals the sum of their clique number and minimum degree. Combining this with Yuan's result, one can deduce that if $G$ is a $2$-connected graph of order $n$, then $c(G)\geq \min\{n,\omega(G)+\delta(G)+1\}$, unless $G$ belongs to certain specified graph classes.