On the least size of a graph with a given degree set -- II

TL;DR

This paper investigates the minimal size of simple graphs with a specified degree set, expanding previous results and providing exact solutions for certain cases along with approximation methods for general degree sets.

Contribution

It extends prior work by determining the least size of graphs with a given degree set in new cases and introduces an approximation algorithm for general degree sets.

Findings

Exact minimal sizes for specific degree set conditions.

A graph construction within a factor of 1+2/d₁ of the optimal size.

General bounds and approximation ratios for graph sizes with given degree sets.

Abstract

The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\mathscr D$ is $1+\max \mathscr D$. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set $\mathscr D$. We expand on their results, and determine the least size of graphs with degree set $\mathscr D$ when (i) $\min \mathscr D \mid d$ for each $d \in \mathscr D$; (ii) $\min \mathscr D=2$; (iii) $\mathscr D=\{m,m+1,\ldots,n\}$. In addition, given any $\mathscr D$, we produce a graph $G$ whose size is within $\min \mathscr D$ of the optimal size, giving a $\big(1+\frac{2}{d_1+1})$-approximation, where $d_1=\max \mathscr D$.