Two sufficient conditions for a graphic sequence to have a realization with prescribed clique size

A graphic sequence π=(d1,d2,…,dn) is said to be potentially Kr+1-graphic, if π has a realization G containing Kr+1, a clique of r+1 vertices, as a subgraph. In this paper, we give two simple sufficient conditions for a graphic sequence π=(d1,d2,…,dn) to be potentially Kr+1-graphic. We also show that the two sufficient conditions imply a theorem due to Rao [An Erdös-Gallai type result on the clique number of a realization of a degree sequence unpublished.], a theorem due to Li et al [The Erdös-Jacobson-Lehel conjecture on potentially Pk-graphic sequences is true, Sci. China Ser. A 41 (1998) 510-520.], the Erdös-Jacobson-Lehel conjecture on σ(Kr+1,n) which was confirmed (see [Potentially G-graphical degree sequences, in: Y. Alavi et al. (Eds.), Combinatorics, Graph Theory, and Algorithms, vol. 1, New Issues Press, Kalamazoo Michigan, 1999, pp. 451-460; The smallest degree sum that yields potentially Pk-graphic sequences, J. Graph Theory 29 (1998) 63-72; An extremal problem on the potentially Pk-graphic sequence, Discrete Math. 212 (2000) 223-231; The Erdös-Jacobson-Lehel conjecture on potentially Pk-graphic sequences is true, Sci. China Ser. A 41 (1998) 510-520.]) and the Yin-Li-Mao conjecture on σ(Kr+1-e,n) [An extremal problem on the potentially Kr+1-e-graphic sequences, Ars Combin. 74 (2005) 151-159.], where Kr+1-e is a graph obtained by deleting one edge from Kr+1.