An extremal problem on potentially K p,1,1-graphic sequences

A sequence S is potentially K p,1,1 graphical if it has a realization containing a K p,1,1 as a subgraph, where K p,1,1 is a complete 3-partite graph with partition sizes p,1,1. Let σ(K p,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S)≥ σ(K p,1,1, n) is potentially K p,1,1 graphical. In this paper, we prove that σ (K p,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. AMS Subject Classifications: 05C07, 05C35