Degree powers in graphs: The Erdo″s-Stone theorem
Let 1 ≤ p ≤ r + 1, with r ≥ 2 an integer, and let G be a graph of order n. Let d(v) denote the degree of a vertex v ∈ V(G). We show that if ∑ v∈n V (G)d p (v) > (1-1/r) pn p+1, then G has more than 1/2 6r(r+1)r rn r-1 (r + 1)-cliques sharing a common edge. From this we deduce that if ∑ v∈n V (G)d p (v) > (1-1/r) pn p+1+C, then G contains more than C/ p2 6r(r+1) +1 rrn r-p cliques of order r + 1. In turn, this statement is used to strengthen the Erdo's-Stone theorem by using ∑v∈V(G) d p(v) instead of the number of edges. © 2012 Cambridge University Press.
Combinatorics Probability and Computing
Bollobás, B., & Nikiforov, V. (2012). Degree powers in graphs: The Erdo″s-Stone theorem. Combinatorics Probability and Computing, 21 (1-2), 89-105. https://doi.org/10.1017/S0963548311000654