A generalization of a conjecture due to Erdős, Jacobson and Lehel

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 K m + 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 = ( d 1 , d 2 , … , d n ) of nonnegative integers is r -graphic if it is realizable by an r -graph on n vertices. Let σ ( K m + 1 ( r ) , n ) be the smallest even integer such that each n -term r -graphic sequence with term sum of at least σ ( K m + 1 ( r ) , n ) is realizable by an r -graph containing K m + 1 ( r ) as a subgraph. In this paper, we determine the value of σ ( K m + 1 ( r ) , n ) for sufficiently large n , which generalizes a conjecture due to Erdős, Jacobson and Lehel.