Cliques in the union of C₄-free graphs

الأصل · EN

Let B and R be two simple graphs with vertex set V, and let G(B,R) be the simple graph with vertex set V, in which two vertices are adjacent if they are adjacent in at least one of B and R. We prove that if B and R are two C₄-free graphs on the same vertex set V and G(B,R) is the complete graph, then there exists an B-clique X, an R-clique Y and a clique Z in B and R, such that V=X∪ Y∪ Z. Further, if x∈ Z then x is one of the vertices of some double C₅ in G(B,R). In particular, if also G(B,R) does not contains a double C₅, then V is obedient. We obtain that if B and R are C₄-free graphs then ω(G(B,R))≤ ω(B)+ω(R)+1/2(ω(B),ω(R)) and ω(G(B,R))≤ ω(B)+ω(R)+ω(H(B,R)) where H(B,R) is the simple graph with vertex set V, in which two vertices are adjacent if they are adjacent in B and R.