An Extremal Problem On Potentially Kr+1-H-graphic Sequences

A sequence S is potentially Kp,1,1 graphical if it has a realization containing a Kp,1,1 as a subgraph, where Kp,1,1 is a complete 3-partite graph with partition sizes p, 1, 1. Let σ(Kp,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Kp,1,1, n) is potentially Kp,1,1 graphical. In this paper, we prove that σ(Kp,1,1, n) ≥ 2[((p + 1)(n − 1) + 2)/2] for n ≥ p + 2. We conjecture that equality holds for n ≥ 2p + 4. We prove that this conjecture is true for p = 3.