{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:19Z","timestamp":1753893799856,"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>Motivated by a problem in theoretical computer science suggested by Wigderson, Alon and Ben-Eliezer studied the following extremal problem systematically one decade ago. Given a graph $H$, let $C(n,H)$ be the minimum number $k$ such that the following holds. There are $n$ colorings of $E(K_{n})$ with $k$ colors, each associated with one of the vertices of $K_{n}$, such that for every copy $T$ of $H$ in $K_{n}$, at least one of the colorings that are associated with $V(T)$ assigns distinct colors to all the edges of $E(T)$. In this paper, we obtain several new results in this problem including:\r\n\r\nFor paths of short length, we show that $C(n,P_{4})=\\Omega(n^{1\/5})$ and $C(n,P_{t})=\\Omega(n^{1\/3})$ with $t\\in\\{5,6\\}$, which significantly improve the previously known lower bounds $(\\log{n})^{\\Omega(1)}$.\r\nWe make progress on the problem of Alon and Ben-Eliezer about complete graphs, more precisely, we show that $C(n,K_{r})=\\Omega(n^{2\/3})$ when $r\\geqslant 8$, and $C(n,K_{s,t})=\\Omega(n^{2\/3})$ for all $t\\geqslant s\\geqslant 7$.\r\nWhen $H$ is a star with at least $4$ leaves, a matching of size at least $4$, or a path of length at least $7$, we give a new lower bound for $C(n,H)$. We also show that for any graph $H$ with at least $6$ edges, $C(n,H)$ is polynomial in $n$. All of these improve the corresponding results obtained by Alon and Ben-Eliezer.\r\n<\/jats:p>","DOI":"10.37236\/11911","type":"journal-article","created":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T10:04:09Z","timestamp":1720778649000},"source":"Crossref","is-referenced-by-count":0,"title":["Local Rainbow Colorings for Various Graphs"],"prefix":"10.37236","volume":"31","author":[{"given":"Xinbu","family":"Cheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zixiang","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,6,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p55\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p55\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T10:04:10Z","timestamp":1720778650000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i2p55"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,28]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/11911","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,6,28]]},"article-number":"P2.55"}}