{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:21:15Z","timestamp":1759638075340,"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>Let $G$ be a simple graph and $\\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\\Delta(T)\\geq\\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\\Delta(T) \\le \\Big\\lceil\\frac{n}{2}\\Big\\rceil$, then there exists a harmonious colouring of $T$ with $\\Big\\lceil \\frac{n}{2}\\Big \\rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\\leq \\Big\\lceil \\frac{n}{2} \\Big\\rceil +1$.<\/jats:p>","DOI":"10.37236\/9","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:35:17Z","timestamp":1578710117000},"source":"Crossref","is-referenced-by-count":6,"title":["On Harmonious Colouring of Trees"],"prefix":"10.37236","volume":"19","author":[{"given":"A.","family":"Aflaki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Akbari","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K.J.","family":"Edwards","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D.S.","family":"Eskandani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Jamaali","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H.","family":"Ravanbod","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,1,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T17:57:45Z","timestamp":1579283865000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/9","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,1,6]]},"article-number":"P3"}}