Cliques in the Union of C₄ -Free Graphs

Let B and R be two simple C₄-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)+12min(ω(B),ω(R))andω(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).