Conditions for rr-graphic sequences to be potentially Km+1(r)-graphic

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 said to be rr-graphic if it is realizable by an rr-graph on nn vertices. An rr-graphic sequence ππ is said to be potentially Km+1(r)-graphic if it has a realization containing Km+1(r) as a subgraph. In this paper, some conditions for rr-graphic sequences to be potentially Km+1(r)-graphic are given. These are generalizations from 11-graphs to rr-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].