A General Lower Bound for Potentially H-Graphic Sequences

We consider an extremal problem for graphs as introduced by Erdős, Jacobson and Lehel in [7]. Let π be an n-element graphic sequence. Let H be a graph. We wish to determine the smallest m such that any nterm graphic sequence π whose terms sum to at least m has some realization containing H as a subgraph. Denote this value m by σ(H, n). For an arbitrarily chosen H, we give a construction that yields the best known lower bound on σ(H, n). Furthermore, we conjecture that our construction is best possible, as this is the case for all choices of H where σ(H, n) is currently known. We support this conjecture by examining the class of graphs with independence number two and, in this case, determine σ(H, n) precisely. We conclude with a brief discussion of a connection between potentially H-graphic sequences and H-saturated graphs of minimum size.