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

TL;DR

This paper addresses the circumference problem for 2-connected graphs with given minimum degree and clique number. It extends Yuan's lower bound by proving a stability version of Erdős–Gallai: for a 2-connected graph of order $n$, the circumference satisfies $c(G) \ge \min\{n, \omega+\delta+1\}$ except for a finite list of exceptional graphs, which are precisely categorized into explicit families such as $H(n,\omega,\delta)$, $Z(n,\omega,\delta)$, and the collections $H_1$, $H_2$, $H_3$, $H_4$, $G_1$–$G_4$, along with a broader family ${\mathcal G}$. The proof combines classic results (Pósa–Pósa and Erdős–Gallai) with a detailed structural analysis of longest $(H,T)$-paths and a rooted-graph framework, yielding a complete extremal classification for the circumference in terms of $\omega$ and $\delta$. The main contribution is both a sharp bound and a comprehensive description of all extremal (non-hamiltonian) 2-connected graphs achieving the threshold, thereby enriching the understanding of cycle structure in dense graphs and guiding potential extensions to related extremal problems.

Abstract

The circumference denoted by $c(G)$ of a graph $G$ is the length of its longest cycle. Let $δ(G)$ and $ω(G)$ 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 $c(G)\geq \min\{n,ω(G)+δ(G)\}$ 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 $c(G)\geq \min\{n,ω(G)+δ(G)+1\}$, unless $G$ belongs to certain specified graph classes.

Figures (6)

• Figure 1: The vertices in each gray ellipse induce a complete graph.
• Figure 2: The vertices in each gray ellipse induce a complete graph.
• Figure 3: The vertices in each gray ellipse induce a complete graph.
• Figure 4: The red dotted line is not necessarily present.
• Figure 5: The red path is a $(y_1, y_2)$-path we identified.

Theorems & Definitions (39)

• Theorem 1.1: Yuan Yuan
• Theorem 1.8
• Lemma 3.1: P$\mathrm{\acute{o}}$sa Po
• Lemma 3.2: Erdős and Gallai EG
• Lemma 3.3
• proof
• proof
• proof
• Remark 3.4
• Lemma 4.1