{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T15:16:47Z","timestamp":1776179807360,"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>\r\nGiven a system $(G_1, \\ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \\ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\\bigcup_{j=1}^{m}I_j = V$. For a class $\\mathcal{G}$ of graphs, let $m_{\\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $\\Omega(\\log\\log d) \\le m_\\mathcal{T}(d) \\le O(\\log d)$ and $\\Omega(\\log d)\\le m_\\mathcal{B}(d) \\le O(d\/\\log d)$, where $\\mathcal{T}$ is the class of trees and $\\mathcal{B}$ is the class of bipartite graphs.\r\n<\/jats:p>","DOI":"10.37236\/8111","type":"journal-article","created":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T11:21:57Z","timestamp":1581074517000},"source":"Crossref","is-referenced-by-count":4,"title":["Cooperative Colorings of Trees and of Bipartite Graphs"],"prefix":"10.37236","volume":"27","author":[{"given":"Ron","family":"Aharoni","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eli","family":"Berger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maria","family":"Chudnovsky","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fr\u00e9d\u00e9ric","family":"Havet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zilin","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,2,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p41\/8027","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p41\/8027","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T11:21:58Z","timestamp":1581074518000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8111","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,7]]},"article-number":"P1.41"}}