Let $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is contained in a maximal clique with respect to $\subseteq$ (this is an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{\text{MaxCliq}(G)}\MC$ denote the collection of all maximal cliques of $G$.

The max-clique chromatic number of $G$, denoted by $\chi_m(G)$ is defined to be the chromatic number of the hypergraph $(V(G), \MC)$. Clearly, for every graph $G$ we have $\chi_m(G) < \chi(G)$, as every vertex coloring in the graph sense is a coloring in the hypergraph sense.

Note that if $G$ is a triangle-free graph, then $\chi_m(G) = \chi(G)$. On the other hand, for every complete graph $K$ we have $\chi_m(K) = 2$.

Question. Given any integer $n>2$, is there a graph $G$ with $|C| \geq n$ for all $C \in \MC$ and $\chi_m(G) = \chi(G)$?

Let $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is contained in a maximal clique with respect to $\subseteq$ (this is an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{\text{MaxCliq}(G)}\MC$ denote the collection of all maximal cliques of $G$.

The max-clique chromatic number of $G$, denoted by $\chi_m(G)$ is defined to be the chromatic number of the hypergraph $(V(G), \MC)$. Clearly, for every graph $G$ we have $\chi_m(G) \leq \chi(G)$, as every vertex coloring in the graph sense is a coloring in the hypergraph sense.

Note that if $G$ is a triangle-free graph, then $\chi_m(G) = \chi(G)$. On the other hand, for every complete graph $K$ we have $\chi_m(K) = 2$.

Question. Given any integer $n>2$, is there a graph $G$ with $|C| \geq n$ for all $C \in \MC$ and $\chi_m(G) = \chi(G)$?