{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:23Z","timestamp":1753893863278,"version":"3.41.2"},"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>For graph $G$ and integers $a_1 \\ge \\cdots \\ge a_r \\ge 2$, we write $G \\rightarrow (a_1 ,\\cdots ,a_r)^v$\u00a0 if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \\in \\{1, \\cdots, r\\}$. The vertex Folkman number $F_v(a_1 ,\\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \\rightarrow (a_1 ,\\cdots ,a_r)^v$. It is well known that if $G \\rightarrow (a_1 ,\\cdots ,a_r)^v$ then $\\chi(G) \\geq m$, where $m = 1+ \\sum_{i=1}^r (a_i - 1)$. In this paper we study such Folkman graphs $G$ with chromatic number $\\chi(G)=m$, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all $r,s \\ge 2$ there exist $K_{s+1}$-free graphs $G$ such that $G \\rightarrow (s,\\cdots_r,s)^v$ and $G$ has the smallest possible chromatic number $r(s-1)+1$ with respect to this property. Among others we conjecture that for every $s \\ge 2$ there exists a $K_{s+1}$-free graph $G$ on $F_v(s,s;s+1)$ vertices with $\\chi(G)=2s-1$ and $G\\rightarrow (s,s)^v$.<\/jats:p>","DOI":"10.37236\/7862","type":"journal-article","created":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:23Z","timestamp":1599187643000},"source":"Crossref","is-referenced-by-count":0,"title":["Chromatic Vertex Folkman Numbers"],"prefix":"10.37236","volume":"27","author":[{"given":"Xiaodong","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Meilian","family":"Liang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stanis\u0142aw P.","family":"Radziszowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,9,4]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p53\/8171","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p53\/8171","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:23Z","timestamp":1599187643000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,4]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"https:\/\/doi.org\/10.37236\/7862","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,9,4]]},"article-number":"P3.53"}}