On the Approximate Shape of Degree Sequences That Are Not Potentially H-graphic

A sequence of nonnegative integers π is graphic if it is the degree sequence of some graph G. In this case we say that G is a realization of π, and we write π = π(G). A graphic sequence π is potentially H-graphic if there is a realization of π that contains H as a subgraph. Given nonincreasing graphic sequences π1 = (d1, . . . , dn) and π2 = (s1, . . . , sn), we say that π1 majorizes π2 if di ≥ si for all i, 1 ≤ i ≤ n. In 1970, Erdős showed that for any Kr+1-free graph H, there exists an r-partite graph G such that π(G) majorizes π(H). In 2005, Pikhurko and Taraz generalized this notion and showed that for any graph F with chromatic number r+ 1, the degree sequence of an F -free graph is, in an appropriate sense, nearly majorized by the degree sequence of an r-partite graph. In this paper, we give similar results for degree sequences that are not potentially Hgraphic. In particular, there is a graphic sequence π∗(H) such that if π is a graphic sequence that is not potentially H-graphic, then π is close to being majorized by π∗(H). Similar to the role played by complete multipartite graphs in the traditional extremal setting, the sequence π∗(H) asymptotically gives the maximum possible sum of a graphic sequence π that is not potentially H-graphic. AMS 2010 Subject Classification: Primary 05C07; Secondary 05C35