{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:14:34Z","timestamp":1772136874920,"version":"3.50.1"},"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>A graph $G$ is $k$-locally sparse if for each vertex $v \\in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f &gt; 0$ if a graph $G$ of maximum degree $\\Delta$ is $\\Delta^2\/f$-locally-sparse, then $\\chi(G) = O\\left(\\Delta\/\\log f\\right)$. We introduce a more general notion of local sparsity by defining graphs $G$ to be $(k, F)$-locally-sparse for some graph $F$ if for each vertex $v \\in V(G)$ the subgraph induced by the neighborhood of $v$ contains at most $k$ copies of $F$. Employing the R\u00f6dl nibble method, we prove the following generalization of the above result: for every bipartite graph $F$, if $G$ is $(k, F)$-locally-sparse, then $\\chi(G) = O\\left( \\Delta \/\\log\\left(\\Delta k^{-1\/|V(F)|}\\right)\\right)$. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our results also recover the best known bound on $\\chi(G)$ when $G$ is $K_{1, t, t}$-free for $t \\geq 4$, and hold for list and correspondence coloring in the more general so-called \"color-degree\" setting.<\/jats:p>","DOI":"10.37236\/13620","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:50Z","timestamp":1772135030000},"source":"Crossref","is-referenced-by-count":0,"title":["Coloring Locally Sparse Graphs"],"prefix":"10.37236","volume":"33","author":[{"given":"James","family":"Anderson","sequence":"first","affiliation":[]},{"given":"Abhishek","family":"Dhawan","sequence":"additional","affiliation":[]},{"given":"Aiya","family":"Kuchukova","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,2,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:51Z","timestamp":1772135031000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13620","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,13]]},"article-number":"P1.31"}}