{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:23Z","timestamp":1753893803019,"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>In 1974, Erd\u0151s and Rothschild initiated to study the maximum possible number, known as $F(n,r,k)$, of distinct edge-colorings of a graph on $n$ vertices with $r$ colors which contain no monochromatic copy of $K_k$. It is not well understood but a few of non-trivial cases. Recently, Balogh, Liu and Sharifzadeh (2017) introduced an extension of such Erd\u0151s-Rothschild problem: given a function $f(n)$ and a graph $H$, let $RF(n,r,H,f(n))$ be the maximum number of distinct $r$-edge-colorings that an $n$-vertex graph with independence number at most $f(n)$ can have without a monochromatic copy of $H$. In particular, they determined the values of $RF(n,2,K_k,o(n))$ for $k\\ge3$ and $RF(n,3,K_3,o(n))$.\r\nDefine the forest arboricity of $H$, denoted $arb_f(H)$, as the minimum integer $p$ such that $V(H)$ can be partitioned into $\\lceil\\frac{p}{2}\\rceil$ sets $V_1,\\ldots,V_{\\lceil\\frac{p}{2}\\rceil}$ such that $V_i$ spans a forest for each $1\\le i\\le{\\lfloor\\frac{p}{2}\\rfloor}$, and the last class $V_{\\lceil\\frac{p}{2}\\rceil}$ spans an independent set if $p$ is odd. In this paper, we mainly obtain the asymptotic values of $RF(n,r,H,o(n))$ for $r\\in\\{3,4,5\\}$, where $H$ is any graph with $arb_f(H)=3$ and chromatic number $\\chi(H)\\ge3$. As a corollary, we have the asymptotic values of $RF(n,r,H,o(n))$ for $r\\in\\{3,4,5\\}$ when $H$ is an odd cycle, or a book (fan) graph.\u00a0<\/jats:p>","DOI":"10.37236\/12016","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T13:52:09Z","timestamp":1739541129000},"source":"Crossref","is-referenced-by-count":0,"title":["The Number of Edge Colorings with Small Independence Number and No Monochromatic $H$"],"prefix":"10.37236","volume":"32","author":[{"given":"Xinyu","family":"Hu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qizhong","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiaxi","family":"Nie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,2,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p25\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T13:52:10Z","timestamp":1739541130000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i1p25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1,17]]}},"URL":"https:\/\/doi.org\/10.37236\/12016","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,2,14]]},"article-number":"P1.25"}}