{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:05Z","timestamp":1753893785492,"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>Let ${\\cal G}$ be a class of graphs. A $d$-fold grid over ${\\cal G}$ is a graph obtained from a $d$-dimensional rectangular grid of vertices by placing a graph from ${\\cal G}$ on each of the lines parallel to one of the axes.  Thus each vertex belongs to $d$ of these subgraphs.  The class of $d$-fold grids over ${\\cal G}$ is denoted by ${\\cal G}^d$. Let $f({\\cal G};d)=\\max_{G\\in{\\cal G}^d}\\chi(G)$.  If each graph in ${\\cal G}$ is $k$-colorable, then $f({\\cal G};d)\\le k^d$.  We show that this bound is best possible by proving that $f({\\cal G};d)=k^d$ when ${\\cal G}$ is the class of all $k$-colorable graphs.  We also show that  $f({\\cal G};d)\\ge{\\left\\lfloor\\sqrt{{d\\over 6\\log d}}\\right\\rfloor}$  when ${\\cal G}$ is the class of graphs with at most one edge, and $f({\\cal G};d)\\ge {\\left\\lfloor{d\\over 6\\log d}\\right\\rfloor}$ when ${\\cal G}$ is the class of graphs with maximum degree $1$.<\/jats:p>","DOI":"10.37236\/160","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:29:09Z","timestamp":1578716949000},"source":"Crossref","is-referenced-by-count":0,"title":["Chromatic Number for a Generalization of Cartesian Product Graphs"],"prefix":"10.37236","volume":"16","author":[{"given":"Daniel","family":"Kr\u00e1l'","sequence":"first","affiliation":[]},{"given":"Douglas B.","family":"West","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,6,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r71\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r71\/comment","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r71\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:52:45Z","timestamp":1579315965000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,6,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/160","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,6,19]]},"article-number":"R71"}}