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

An rr-graph is a loopless undirected graph in which no two vertices are joined by more than rr edges. An rr-complete graph on m+1m+1 vertices, denoted by Km+1(r), is an rr-graph on m+1m+1 vertices in which each pair of vertices is joined by exactly rr edges. A non-increasing sequence π=(d1,d2,…,dn)π=(d1,d2,…,dn) of nonnegative integers is rr-graphic if it is realizable by an rr-graph on nn vertices. Let σ(Km+1(r),n) be the smallest even integer such that each nn-term rr-graphic sequence with term sum of at least σ(Km+1(r),n) is realizable by an rr-graph containing Km+1(r) as a subgraph. In this paper, we determine the value of σ(Km+1(r),n) for sufficiently large nn, which generalizes a conjecture due to Erdős, Jacobson and Lehel.