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

Jai Moondra, Aditya Sahdev, Amitabha Tripathi

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 $d_1=\max \mathscr D$.

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

Abstract

The degree set of a finite simple graph is the set of distinct degrees of vertices of . A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set is . Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set . We expand on their results, and determine the least size of graphs with degree set when (i) for each ; (ii) ; (iii) . In addition, given any , we produce a graph whose size is within of the optimal size, giving a -approximation, where .

Paper Structure

This paper contains 5 sections, 14 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Basic Results
  3. Two special cases
  4. The case where $\mathscr D$ is an interval
  5. An approximation solution for the general case

Key Result

Theorem 1

(KPW77) For each nonempty finite set $\mathscr D$ of positive integers, there exists a simple graph $G$ for which ${\mathscr D}(G)=\mathscr D$. Moreover, there is always such a graph of order $\Delta+1$, where $\Delta=\max \mathscr D$, and there is no such graph of smaller order.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Lemma 1
  • Theorem 4
  • Corollary 1
  • Theorem 5
  • Theorem 6
  • Proposition 2
  • ...and 5 more