Cliques in the Union of $$C_4$$ 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 $$E(B) \cap E(R)$$ .