{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:33Z","timestamp":1753893813992,"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>Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges must receive different colours. Let $\\pi'_t(d)$ and $\\tau'_t(d)$ be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree $d$. We have that $\\pi'_t(d)$ is at most and at least a non-trivial constant multiple larger than $\\tau'_t(d)$. (We conjecture $\\limsup_{d\\to\\infty}\\pi'_2(d)\/\\tau'_2(d) =9\/4$ in particular.) We prove for odd $t$ the existence of a quantity $g$ depending only on $t$ such that the distance-$t$ chromatic index of any planar multigraph of maximum degree $d$ and girth at least $g$ is at most $\\tau'_t(d)$ if $d$ is sufficiently large. Such a quantity does not exist for even $t$. We also show a related, similar phenomenon for distance vertex-colouring.<\/jats:p>","DOI":"10.37236\/8220","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:12:04Z","timestamp":1578640324000},"source":"Crossref","is-referenced-by-count":0,"title":["Tree-Like Distance Colouring for Planar Graphs of Sufficient Girth"],"prefix":"10.37236","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4219-593X","authenticated-orcid":false,"given":"Ross J.","family":"Kang","sequence":"first","affiliation":[]},{"given":"Willem","family":"Van Loon","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,2,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i1p23\/7782","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i1p23\/7782","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:17:57Z","timestamp":1579234677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i1p23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1,11]]}},"URL":"https:\/\/doi.org\/10.37236\/8220","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,2,22]]},"article-number":"P1.23"}}