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{\bf Andr\'as Gy\'arf\'as and Manouchehr Zaker}
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{\bf On $(\delta, \chi)$-Bounded Families of Graphs}
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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.
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