On the existence of two non-neighboring subgraphs in a graph

hwo subgraphs G r , G_ of a graph G are called non-neighboring if there is no edge v,v_EE(G) with v,EG, and v_EG_ . In general, an arbitrary graph may not contain two non-neighboring subgraphs at all, for example the complete graph . In this paper we raise the following question : Is there a minimal integer, f( n) such that each graph G with Z(G) -f(r, n) and which does not contain a complete subgraph of order r must contain two non-neighboring n-chromatic subgraphs? An upper bound for f (r, 2) follows from a result of S . Wagon [2] . Here we show that it is sufficient to prove the existence of f (r, n) for r-n . More precisely, for a fixed n, an upper bound for .f(r . It), r :it, is given in terms of f(r, n), r=n . The proof is based on the same idea of S . Wagon . From f(3, 3) `=8 we deduce an upper bound for f(r, 3) . Next we muesli ate graphs which do not contain 2K, as an induced subgraph . We say that the two edges v, v_, u, it., of the graph G are independent if the subgraph induced by v,, v .,, it,, u, is 2K,., i .e ., the complement of a chordless 4-cycle . We prove that a vertex-critical 4-chromatic graph G which does not contain two independent edges has order iG'1 __ 13. We also give a lower bound for the maximum degree of a graph without two independent edges .