On the approximate shape of degree sequences that are not potentially $H$-graphic

A sequence of nonnegative integers $π$ is {\it graphic} if it is the degree sequence of some graph $G$. In this case we say that $G$ is a \textit{realization} of $π$, and we write $π=π(G)$. A graphic sequence $π$ is {\it potentially $H$-graphic} if there is a realization of $π$ that contains $H$ as a subgraph. Given nonincreasing graphic sequences $π_1=(d_1,\ldots,d_n)$ and $π_2 = (s_1,\ldots,s_n)$, we say that $π_1$ {\it majorizes} $π_2$ if $d_i \geq s_i$ for all $i$, $1 \leq i \leq n$. In 1970, Erdős showed that for any $K_{r+1}$-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 $H$-graphic. 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.

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