An extremal problem on the potentially Pk-graphic sequences

Abstract

A simple graph G is said to have property P k if it contains a complete subgraph of order k+1 as its subgraph. A nonincreasing sequence π of n nonnegative integers is potentially P k -graphic if it is the degree sequence of a graph of order n with property P k . The degree sum of a graphic sequence π is denoted by σ( π). Moreover we denote by σ( k, n) the smallest degree sum such that every positive graphic sequence π with σ( π)⩾ σ( k, n) is potentially P k -graphic. Erdős et al. (Graph Theory, Combinatorics & Applications, Wiley, New York, 1991, pp. 439–449) conjectured σ( k, n)=( k−1)(2 n− k)+2. In this paper, we determine the values of σ( k, n) for k+1⩽ n⩽2 k+1. We also prove that σ(3, n)=4 n−4 for n⩾8. In other words, the Erdős–Jacobson–Lehel conjecture is true for k=3.