{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T22:40:06Z","timestamp":1757630406755,"version":"3.44.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We relate star colouring of even-degree regular graphs to the notions of locally constrained graph homomorphisms to the oriented line graph $\\vec{L}(K_q)$ of the complete graph $K_q$ and to its underlying undirected graph $L^*(K_q)$. Our results have consequences for locally constrained graph homomorphisms and oriented line graphs in addition to star colouring. We show that $L^*(H)$ is a 2-lift of the line graph $L(H)$ for every graph $H$. Dvo\u0159\u00e1k, Mohar and \u0160\u00e1mal (J. Graph Theory, 2013) proved that for every 3-regular graph $G$, the line graph of $G$ is 4-star colourable if and only if $G$ admits a locally bijective homomorphism to the cube $Q_3$. We generalise this result as follows: for $p\\geq 2$, a $K_{1,p+1}$-free $2p$-regular graph $G$ admits a $(p+2)$-star colouring if and only if $G$ admits a locally bijective homomorphism to $L^*(K_{p+2})$. As a result, if a $K_{p+1}$-free $2p$-regular graph $G$ with $p\\geq 2$ is $(p+2)$-star colourable, then $-2$ and $p-2$ are eigenvalues of $G$. We also prove the following:(i) for $p\\geq 2$, a $2p$-regular graph $G$ admits a $(p+2)$-star colouring if and only if $G$ has an orientation that admits an out-neighbourhood bijective homomorphism to $\\vec{L}(K_{p+2})$; (ii) the line graph of a 3-regular graph $G$ is 4-star colourable if and only if $G$ is bipartite and distance-two 4-colourable; and (iii) it is NP-complete to check whether a planar 4-regular 3-connected graph is 4-star colourable.<\/jats:p>","DOI":"10.37236\/12949","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T17:49:40Z","timestamp":1757526580000},"source":"Crossref","is-referenced-by-count":0,"title":["Star Colouring and Locally Constrained Graph Homomorphisms"],"prefix":"10.37236","volume":"32","author":[{"given":"Cyriac","family":"Antony","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shalu","family":"M. A.","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,9,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i3p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i3p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T17:49:40Z","timestamp":1757526580000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i3p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,5]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/12949","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,5]]},"article-number":"P3.43"}}