On the potentially Pk-graphic sequences

A nonincreasing sequence pi of n nonnegative integers is said to be graphic if it is the degree sequence of a simple graph G of order n and G is called a realization of n. A graph G of order n is said to have property P-k if it contains a clique of size k as a subgraph. An n-term graphic sequence pi is said to be potentially (res. forcibly)P-k-graphic if it has a realization having (res. all its realizations have) property P-k. It is well known that, if t(k-1)(n) is the Turan number, then t(k-1)(n) is the smallest number such that each graph G of order n with edge number epsilon(G)greater than or equal to t(k-1)(n) + 1 has property P-k. The Turan theorem states that t(k-1)(n) = ((n)(2)) - t(n - k + 1 - r)/2, where n = t(k - 1) + r, 0 less than or equal to r < k - 1. In terms of graphic sequences, 2(t(k-1)(n) + 1) is the smallest even number such that each graphic sequence pi = (d(1), d(2) ,..., d(n)) with sigma(pi)= d(1) + d(2) + ... + d(n) greater than or equal to 2(t(k-1)(n)+ 1) is forcibly P-k-graphic. In 1991, Erdos et al. [1] considered a variation of this classical extremal problem: determine the smallest even number sigma(k, n) such that each graphic sequence pi = (d(1), d(2) ,..., d(n)) with d1 greater than or equal to d(2) greater than or equal to ... greater than or equal to d(n) greater than or equal to 1 and sigma(pi) greater than or equal to sigma(k, n) is potentially P-k-graphic. They gave a lower bound of sigma(k, n), i.e., sigma(k, n) greater than or equal to (k - 2)(2n - k + 1)+ 2 and conjectured that the lower bound is the exact value of sigma(k,n). In this paper, we prove the upper bound sigma(k,n)less than or equal to 2n(k - 2) + 2 for n greater than or equal to 2k - 1. (C) 1999 Elsevier Science B.V. All rights reserved.