A note on chromatic roots of clique-theta graphs
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In this paper,the structural characterization of {2K_1 + K_2,C_4}-free graphs was given according to their independence numbers,and the linear upper bound on chromatic number of {2K_1+ K_2,C_4}-free graphs was obtained in terms of their clique number.
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The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.
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