Cliques in the union of graphs

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. For X ⊆ V , we denote by B|X the subgraph of B induced by X; let R|X and G(B,R)|X be defined similarly. A clique in a graph is a set of pairwise adjacent vertices. A subset U ⊆ V is obedient if U is the union of a clique of B and a clique of R. Our first result is that if B has no induced cycles of length four, and R has no induced cycles of length four or five, then every clique of G(B,R) is obedient. This strengthens a previous result of the second author, stating the same when B has no induced C4 and R is chordal. The clique number of a graph is the size of its maximum clique. We say that the pair (B,R) is additive if for every X ⊆ V , the sum of the clique numbers of B|X and R|X is at least the clique number of G(B,R)|X. Our second result is a sufficient condition for additivity of pairs of graphs.