{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:18:35Z","timestamp":1759637915890,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We examine the effect of bounding the diameter for a number of natural and well-studied variants of the\u00a0 COLOURING problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are ACYCLIC COLOURING, STAR COLOURING and INJECTIVE COLOURING. The last problem is also known as $L(1,1)$-LABELLING and we also consider the framework of $L(a,b)$-LABELLING. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most~$d$, ACYCLIC $3$-COLOURING is polynomial-time solvable if $d\\leq 2$ but NP-complete if $d\\geq 4$, and STAR $3$-COLOURING is polynomial-time solvable if $d\\leq 3$ but NP-complete for $d\\geq 8.$ As far as we are aware, STAR $3$-COLOURING is the first problem that exhibits a complexity jump for some $d\\geq 3.$ Our third main result is that $L(1,2)$-LABELLING is NP-complete for graphs of diameter~$2$; we relate the latter problem to a special case of HAMILTONIAN PATH.<\/jats:p>","DOI":"10.37236\/10738","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T12:11:09Z","timestamp":1654258269000},"source":"Crossref","is-referenced-by-count":2,"title":["Acyclic, Star, and Injective Colouring: Bounding the Diameter"],"prefix":"10.37236","volume":"29","author":[{"given":"Christoph","family":"Brause","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Petr","family":"Golovach","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Barnaby","family":"Martin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pascal","family":"Ochem","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dani\u00ebl","family":"Paulusma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siani","family":"Smith","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2022,6,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i2p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i2p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T12:11:09Z","timestamp":1654258269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i2p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,4,8]]}},"URL":"https:\/\/doi.org\/10.37236\/10738","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,6,3]]},"article-number":"P2.43"}}