Jian Hua YIN
An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by Km+1(r), is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d1, d2,..., dn) of nonnegative integers is said to be r-graphic if it is the degree sequence of some r-graph on n vertices. An r-graphic sequence π is said to be potentially (resp. forcibly) Km+1(r)-graphic if π has a realization containing Km+1(r) as a subgraph (resp. every realization of π contains Km+1(r) as a subgraph). Let σ(Km+1(r), n) (resp. τ (Km+1(r), n)) denote the smallest even integer t such that each r-graphic sequence π=(d1,d2,..., dn) with ∑i=1ndi ≥ t is potentially (resp. forcibly) Km+1(r)-graphic. Clearly, σ(Km+1(r), n) is an extension from 1-graph to r-graph of a conjecture due to Erdös et al and τ (Km+1(r), n) is an extension from 1-graph to r-graph of the classical Turán's theorem. In this paper, we give two simple sufficient conditions for an r-graphic sequence to be potentially Km+1(r)-graphic, which imply two main results due to Yin and Li (Discrete Math., 2005, 301: 218-227) and the value of σ(Km+1(r), n) for n ≥ max{m2+3m+1-[m2+m/r], 2m+1+[m/r]}. Moreover, we also determine the value of τ(Km+1(r), n) for n ≥ m+1.