The smallest degree sum that yields potentially Pk-graphical sequences

A simple graphG is said to have property Pk if it contains a complete subgraph of order k + 1, and a sequence π is potentially Pk-graphical if it has a realization having property Pk. Let σ(k, n) denote the smallest degree sum such that every n-term graphical sequence π without zero terms and with degree sum σ(π) ≥ σ(k, n) is potentially Pk-graphical. Erdös, Jacobson, and Lehel [Graph Theory, 1991, 439--449] conjectured that σ(k, n) = (k − 1)(2n − k) + 2. In this article, we prove that the conjecture is true for k = 4 and n ≥ 10. c © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 63–72, 1998