{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:26Z","timestamp":1753893806184,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We say that a graph $G$ has an odd $K_4$-subdivision if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\\ell\\geq 2$, let $\\mathcal{G}_{\\ell}$ denote the family of graphs which have girth $2\\ell+1$ and have no odd hole with length greater than $2\\ell+1$. Wu, Xu and Xu conjectured that every graph in $\\bigcup_{\\ell\\geq2}\\mathcal{G}_{\\ell}$ is $3$-colorable. Recently, Chudnovsky et al. and Wu et al., respectively, proved that every graph in $\\mathcal{G}_2$ and $\\mathcal{G}_3$ is $3$-colorable. In this paper, we prove that no $4$-vertex-critical graph in $\\bigcup_{\\ell\\geq5}\\mathcal{G}_{\\ell}$ has an odd $K_4$-subdivision. Using this result, Chen proved that all graphs in $\\bigcup_{\\ell\\geq5}\\mathcal{G}_{\\ell}$ are $3$-colorable.\u00a0<\/jats:p>","DOI":"10.37236\/12135","type":"journal-article","created":{"date-parts":[[2024,2,22]],"date-time":"2024-02-22T16:18:14Z","timestamp":1708618694000},"source":"Crossref","is-referenced-by-count":1,"title":["Graphs with Girth $2\\ell+1$ and Without Longer Odd Holes that Contain an Odd $K_4$-Subdivision"],"prefix":"10.37236","volume":"31","author":[{"given":"Rong","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yidong","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,2,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p45\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,22]],"date-time":"2024-02-22T16:18:14Z","timestamp":1708618694000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i1p45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/12135","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,2,23]]},"article-number":"P1.45"}}