{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:11:10Z","timestamp":1758823870950,"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>In this paper, we introduce a new variation of list-colorings. For a graph $G$\u00a0 and for a given nonnegative integer $t$, a\u00a0$t$-common list assignment\u00a0of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\\in V(G)$. The $t$-common list chromatic number\u00a0of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \\ge k$ for every vertex $v\\in V(G)$. We show that for all positive integers $k, \\ell$ with $2 \\le k \\le \\ell$ and for any positive integers $i_1 , i_2, \\ldots, i_{k-2}$ with $k \\le i_{k-2} \\le \\cdots \\le i_1 \\le \\ell$, there exists a graph $G$ such that $\\chi(G)= k$, $ch(G) =\u00a0 \\ell$ and $ch_t(G) = i_t$ for every $t=1, \\ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.<\/jats:p>","DOI":"10.37236\/6738","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:50:06Z","timestamp":1578653406000},"source":"Crossref","is-referenced-by-count":4,"title":["On $t$-Common List-Colorings"],"prefix":"10.37236","volume":"24","author":[{"given":"Hojin","family":"Choi","sequence":"first","affiliation":[]},{"given":"Young Soo","family":"Kwon","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,8,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:50:08Z","timestamp":1579218608000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,11]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6738","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,8,11]]},"article-number":"P3.32"}}