On the potentially Pk-graphic sequences

Abstract

A nonincreasing sequence π 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 π. 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 π 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 Turán number, then t k−1 ( n) is the smallest number such that each graph G of order n with edge number ε( G) ⩾ t k−1 ( n) + 1 has property P k . The Turán theorem states that t k−1 ( n) = ( 2 n ) − t( n − k + 1 − r)/2, where n = t( k − 1) + r, 0 ⩽ r < k − 1. In terms of graphic sequences, 2( t k − 1 ( n) + 1) is the smallest even number such that each graphic sequence π = ( d 1, d 2,…, d n ) with σ( π) = d 1, + d 2 + … + d n ⩾ 2( t k−1 ( n) + 1) is forcibly P k-graphic. In 1991, Erdös et al. [1] considered a variation of this classical extremal problem: determine the smallest even number σ( k, n) such that each graphic sequence π = ( d 1, d 2, …, d n ) with d 1 ⩾ d 2 ⩾ … ⩾ d n ⩾ 1 and σ( π) ⩾ σ ( k, n) is potentially P k - graphic. They gave a lower bound of σ( k, n), i.e., σ( k, n) ⩾ ( k − 2)(2 n − k + 1) + 2 and conjectured that the lower bound is the exact value of σ( k, n). In this paper, we prove the upper bound σ( k, n) ⩽ 2 n( k − 2) + 2 for n ⩾ 2 k − 1.