A neighborhood condition which implies the existence of a complete multipartite subgraph

Abstract

Given a graph G and uϵV( G), the neighborhood N( u)={ uϵV( G)| uvϵE( G)}. We define NC k ( G)= min|∪ N( u i )| where the minimum is taken over all k independent sets { u 1… u k } of vertices in V( G). We shall show that if G is a graph of order n that satisfies the neighborhood condition NC k(G) > d−2 d−1 n+cn 1− 1 r for some real number c= c( m, d, k, r) then for sufficiently large n, G contains at least one copy of a K( r, m… m d−1 ) where m i = m for each i and r⩾ m. When r=1, 2 or 3, this result is best possible.