A sufficient condition for pancyclic graphs

TL;DR

This paper establishes a new sufficient condition for 2-connected graphs to be pancyclic, specifically for $[4,2]$-graphs of order at least 7, and explores properties of triangle-free $[p+2,p]$-graphs.

Contribution

It proves that all 2-connected $[4,2]$-graphs with at least 7 vertices are pancyclic, strengthening previous results and analyzing triangle-free cases for general $p$.

Findings

Every 2-connected $[4,2]$-graph of order ≥7 is pancyclic.

Existence of 2-connected $[4,2]$-graphs not satisfying Chvátal-Erdős condition.

Characterization of triangle-free $[p+2,p]$-graphs for general $p$.

Abstract

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t.$ We prove that every $2$-connected $[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing results. There are $2$-connected $[4,2]$-graphs which do not satisfy the Chv\'{a}tal-Erd\H{o}s condition. We also determine the triangle-free graphs among $[p+2,p]$-graphs for a general $p.$