On the potentially Pk - graphic sequences 1 Jiong - Sheng

A nonincreasing sequence n 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, if it contains a clique of size k as a subgraph. An n-term graphic sequence n is said to be potentially (res. forcibly)Pk-graphic if it has a realization having (res. all its realizations have) property Pk. It is well known that, if tk L(n) is the Tur~in number, then tk-~(n) is the smallest number such that each graph G of order n with edge number g(G)>~t,_l(n) + 1 has property P,. The Tur~in theorem states that t , i (n) = ('~) t(n k + 1 r)/2, where n = t(k 1) + r,0 ~< r < k 1. In terms of graphic sequences, 2(tk-i(n) + 1) is the smallest even number such that each graphic sequence n = (dl ,d2, . . . ,d , , ) with tr(n) = dl + d2 + . . . + d,, >t 2 ( t , l ( n ) + 1) is forcibly Pk-graphic. In 1991, Erd6s et al. [1] considered a variation of this classical extremal problem: determine the smallest even number a(k,n) such that each graphic sequence n = (d l ,d2 , . . . ,dn) with dl >~d2 ~> . . ~>dn t> 1 and a(n)>~a(k,n) is potentially P k graphic. They gave a lower bound of a(k,n), i.e., a(k ,n )> . . . ( k -2 ) (2n-k+ 1)+2 and conjectured that the lower bound is the exact value of a(k,n). In this paper, we prove the upper bound a(k,n)<-..2n(k 2) + 2 for n>j2k 1. @ 1999 Elsevier Science B.V. All rights reserved