The sum of degrees in cliques

Abstract

For every graph G, let Δr (G) = max{∑ uεRd (u) : R is an r-clique of G} and let Δr (n, m) be the minimum of Δr (G) taken over all graphs of order n and size m. Write tr (n) for the size of the r-chromatic Turán graph of order n. Improving earlier results of Edwards and Faudree, we show that for every r ≥ 2; if m ≥ tr (n), then Δr (n, m) ≥ 2rm/n; as conjectured by Bollobás and Erdos. It is known that inequality (1) fails for m < tr (n). However, we show that for every ε > 0, there is δ > 0 such that if m > tr (n) - δn2 then Δr (n, m) ≥ (1 - ε) 2rm/n.