{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:40Z","timestamp":1753893820145,"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>A family ${\\mathcal{F}}$ of graphs is said to be $(\\delta,\\chi)$-bounded if there exists a function $f(x)$ satisfying $f(x)\\rightarrow \\infty$ as $x\\rightarrow \\infty$, such that for any graph $G$ from the family, one has $f(\\delta(G))\\leq \\chi(G)$, where $\\delta(G)$ and $\\chi(G)$ denotes the minimum degree and chromatic number of $G$, respectively. Also for any set $\\{H_1, H_2, \\ldots, H_k\\}$ of graphs by $Forb(H_1, H_2, \\ldots, H_k)$ we mean the class of graphs that contain no $H_i$ as an induced subgraph for any $i=1, \\ldots, k$. In this paper we first answer affirmatively the question raised by the second author by showing that for any tree $T$ and positive integer $\\ell$, $Forb(T, K_{\\ell, \\ell})$ is a $(\\delta, \\chi)$-bounded family. Then we obtain a necessary and sufficient condition for $Forb(H_1, H_2, \\ldots, H_k)$ to be a $(\\delta, \\chi)$-bounded family, where $\\{H_1, H_2, \\ldots, H_k\\}$ is any given set of graphs.  Next we study $(\\delta, \\chi)$-boundedness of $Forb({\\mathcal{C}})$ where ${\\mathcal{C}}$ is an infinite collection of graphs. We show that for any positive integer $\\ell$, $Forb(K_{\\ell,\\ell}, C_6, C_8, \\ldots)$ is $(\\delta, \\chi)$-bounded. Finally we show a similar result when ${\\mathcal{C}}$ is a collection consisting of unicyclic graphs.<\/jats:p>","DOI":"10.37236\/595","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:47:39Z","timestamp":1578696459000},"source":"Crossref","is-referenced-by-count":2,"title":["On $(\\delta, \\chi)$-Bounded Families of Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"Andr\u00e1s","family":"Gy\u00e1rf\u00e1s","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manouchehr","family":"Zaker","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,5,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p108\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p108\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:12:13Z","timestamp":1579284733000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,5,8]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/595","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,5,8]]},"article-number":"P108"}}