excluded subgraph

Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family ${\mathcal F}$ of graphs is $\chi$-\emph{bounded} if there exists a function $f: {\mathbb N} \rightarrow {\mathbb N}$ so that every $G \in {\mathcal F}$ satisfies $\chi(G) \le f (\omega(G))$.

\begin{conjecture} For every fixed tree $T$, the family of graphs with no induced subgraph isomorphic to $T$ is $\chi$-bounded. \end{conjecture}

Keywords: chi-bounded; coloring; excluded subgraph; tree

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