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

TL;DR

This paper characterizes 2-connected graphs with longest cycles equal to the sum of their clique number and minimum degree, extending previous bounds and identifying specific graph classes where the circumference is slightly larger.

Contribution

It provides a stability result for the Erdős-Gallai theorem and characterizes all 2-connected non-hamiltonian graphs with circumference equal to the sum of clique number and minimum degree.

Findings

Characterization of 2-connected non-hamiltonian graphs with specific circumference

Extension of Yuan's lower bound on circumference

Identification of graph classes where circumference exceeds the sum by one

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.