{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:50:03Z","timestamp":1759063803539,"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>It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if $G$ is a connected graph distinct from $C_7$, then there is a $\\chi(G)$-coloring of $G$ in which every vertex $v\\in V(G)$ is an initial vertex of a path $P$ with $\\chi(G)$ vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs.  Electron. J. Combin. 18(1): P17, 9pp, 2011] this was proved with $\\lfloor\\frac{\\chi(G)}{2} \\rfloor $ vertices instead of $\\chi(G)$ vertices. We strengthen this to $\\chi(G)-1$ vertices. We also prove that every connected graph with at least one edge has a proper $k$-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color $i+1$ (mod $k$). $C_5$ shows that $k$ may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the $k$-coloring exists for every $k \\geq \\chi(G)$. In fact, the $k$-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by $1$ (mod $k$) along each edge. The method is based on the circular chromatic number $\\chi_c(G)$. In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number.<\/jats:p>","DOI":"10.37236\/573","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:51:15Z","timestamp":1578696675000},"source":"Crossref","is-referenced-by-count":5,"title":["Rainbow Paths with Prescribed Ends"],"prefix":"10.37236","volume":"18","author":[{"given":"Meysam","family":"Alishahi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ali","family":"Taherkhani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carsten","family":"Thomassen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,4,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p86\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p86\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:14:14Z","timestamp":1579284854000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p86"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/573","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,4,14]]},"article-number":"P86"}}