Cliques in the Union of C_4 -Free Graphs

Let B and R be two simple C_4 -free graphs with the same vertex set V , and let B ∨ R be the simple graph with vertex set V and edge set E(B) ∪ E(R) . We prove that if B ∨ 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∪ Y∪ Z . For general B and R , not necessarily forming together a complete graph, we obtain that ω (B ∨ R)≤ ω (B)+ω (R)+1/2min (ω (B),ω (R)) and ω (B ∨ R)≤ ω (B)+ω (R)+ω (B ∧ R ) where B ∧ R is the simple graph with vertex set V and edge set E(B) ∩ E(R) .