On the Chromatic Number of H-Free Graphs of Large Minimum Degree

The problem of determining the chromatic number of H-free graphs has been well studied, with particular attention to Kr-free graphs with large minimum degree. Recent progress has been made for triangle-free graphs on n vertices with minimum degree larger than n/3. In this paper, we determine the family of r-colorable graphs $${\mathcal{H}_r}$$, such that if $${H \in \mathcal{H}_r}$$, there exists a constant C < (r − 2)/(r − 1) such that any H-free graph G on n vertices with δ(G) > Cn has chromatic number bounded above by a function dependent only on H and C. A value of C < (r − 2)/(r − 1) is given for every $${H \in \mathcal{H}_r}$$, with particular attention to the case when χ(H) = 3.

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