On the Sum Neccesary to Ensure that a Degree Sequence is Potentially H-Graphic

TL;DR

This paper determines the asymptotic minimum degree sum needed to ensure a graphic sequence can realize a graph containing a specific subgraph H, extending classical extremal graph theory results.

Contribution

It provides an asymptotic solution for the minimum sum (H,n) for all graphs H, generalizing previous results and connecting to the Erd
53s-Stone-Simonovits theorem.

Findings

Asymptotic value of (H,n) for all H determined

Extends extremal graph theory to degree sequence problems

Connects potential degree sequences with classical Ture1n problems

Abstract

A sequence of nonnegative integers \pi =(d_1,d_2,...,d_n) is graphic if there is a (simple) graph G with degree sequence \pi. In this case, G is said to realize or be a realization of \pi. Degree sequence results in the literature generally fall into two classes: forcible problems, in which all realizations of a graphic sequence must have a given property, and potential problems, in which at least one realization of \pi must have the given property. Given a graph H, a graphic sequence \pi is potentially H-graphic if there is some realization of \pi that contains H as a subgraph. In 1991, Erd\H{o}s, Jacobson and Lehel posed the following question: Determine the minimum integer \sigma(H,n) such that every n-term graphic sequence with sum at least \sigma(H,n) is potentially H-graphic. As the sum of the terms of \pi is twice the number of edges in any realization of \pi, the Erd\H{o}s-Jacobson-Lehel problem can be viewed as a potential degree sequence relaxation of the (forcible) Tur\'{a}n problem, wherein one wishes to determine the maximum number of edges in a graph that contains no copy of H. While the exact value of \sigma(H,n) has been determined for a number of specific classes of graphs (including cliques, cycles, complete bigraphs and others), very little is known about the parameter for arbitrary H. In this paper, we determine \sigma(H,n) asymptotically for all H, thereby providing an Erd\H{o}s-Stone-Simonovits-type theorem for the Erd\H{o}s-Jacobson-Lehel problem.