{"title":"A sufficient condition for pancyclic graphs","authors":"Xingzhi Zhan","doi":"arxiv-2409.11716","DOIUrl":null,"url":null,"abstract":"A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of\norder $s$ has size at least $t.$ We prove that every $2$-connected\n$[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing\nresults. There are $2$-connected $[4,2]$-graphs which do not satisfy the\nChv\\'{a}tal-Erd\\H{o}s condition. We also determine the triangle-free graphs\namong $[p+2,p]$-graphs for a general $p.$","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11716","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 $p.$
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