A General Lower Bound for PotentiallyH-Graphic Sequences

Abstract

We consider a variation of the classical Turán-type extremal problem as introduced by Erdős, Jacobson, and Lehel in [Graphs realizing the same degree sequences and their respective clique numbers, in Graph Theory, Combinatorics, and Applications, Vol. 1, Wiley, New York, 1991, pp. 439–449]. Let $\pi$ be an n-element graphic sequence and $\sigma(\pi)$ be the sum of the terms in $\pi$, that is, the degree sum. Let H be a graph. We wish to determine the smallest m such that any n-term graphic sequence $\pi$ having $\sigma(\pi)\geq m$ has some realization containing H as a subgraph. Denote this value m by $\sigma(H,n)$. For an arbitrarily chosen H, we construct a graphic sequence $\pi^*(H,n)$ such that $\sigma(\pi^*(H,n))+2\le\sigma(H,n)$. Furthermore, we conjecture that equality holds in general, as this is the case for all choices of H where $\sigma(H,n)$ is currently known. We support this conjecture by examining those graphs that are the complement of triangle-free graphs and showing that the conjecture holds despite the wide variety of structure in this class. We will conclude with a brief discussion of a connection between potentially H-graphic sequences and H-saturated graphs of minimum size.