Cliques in the Union of $C_4$-Free Graphs

Abstract

Let B and R be two simple $C_4$-free graphs with the same vertex set V, and let $B \vee R$ be the simple graph with vertex set V and edge set $E(B) \cup E(R)$. We prove that if $B \vee R$ is a complete graph, then there exists a B-clique X, an R-clique Y and a set Z which is a clique both in B and in R, such that $V=X\cup Y\cup Z$. For general B and R, not necessarily forming together a complete graph, we obtain that

$$\begin{aligned} \omega (B \vee R)\le & {} \omega (B)+\omega (R)+\frac{1}{2}\min (\omega (B),\omega (R))\\&\hbox {and}\\ \omega (B \vee R)\le & {} \omega (B)+\omega (R)+\omega (B \wedge R ) \end{aligned}$$

where $B \wedge R$ is the simple graph with vertex set V and edge set $E(B) \cap E(R)$.