Chromatic number of Cartesian sum of two graphs

PROOF. If G is a graph, we denote by V(G) the set of vertices of G and by E(G) the set of edges of G. We say that a subset S C V(G) is independent if for any a, b in S, (a, b) ErE(G). If S is a set, we denote by I SI the number of elements in S. Now, let SiC V(Gi) (i= 1, 2) be independent sets such that Sil =3(Gi). If S= {ailDa2: a,ES1 and a2 S2 }, then S is an independent set of G1 E G2. Therefore, 3(GC1 @ G2) > I sI =3(G1)3(G2). Suppose TC V(G1CG2) is an independent set such that I TJ ==13(G1 ED G2). For a G V(Gi), let T(a) ={ b G V(G2) : a0b ET}. Then for each a G V(G1), T(a) is an independent set of G2. Therefore, for each aE V(G1), I T(a) I ?fl(G2). Clearly, I TJ = ZaEV(Gi) I T(a) I. But I T(a) I =0 except for those a in an independent set of G1. Hence j3(GC1 E G2) = 1 TJ <l(GC1)f3(G2). This shows that fl(GC1 C G2) =0(GC1)i(G2). LEMMA 2. K(G1)K(G2) K(CK(G1GC2) .