Conditions for [formula omitted]-graphic sequences to be potentially [formula omitted]-graphic

Abstract

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 said to be r -graphic if it is realizable by an r -graph on n vertices. An r -graphic sequence π is said to be potentially K m + 1 ( r ) -graphic if it has a realization containing K m + 1 ( r ) as a subgraph. In this paper, some conditions for r -graphic sequences to be potentially K m + 1 ( r ) -graphic are given. These are generalizations from 1 -graphs to r -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(1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph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].