Irregular Subgraphs

TL;DR

This paper explores conjectures about the existence of spanning irregular subgraphs with balanced degree distributions in regular and minimum degree graphs, providing asymptotic results for large graphs.

Contribution

It introduces two conjectures on spanning irregular subgraphs and proves asymptotic relaxations for large graphs with specific degree conditions.

Findings

Asymptotic existence of spanning subgraphs with balanced degrees in regular graphs

Spanning subgraphs with limited degree repetition in graphs with minimum degree

Results hold when the number of vertices is large and degree conditions are met

Abstract

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$. The second is that every graph on $n$ vertices with minimum degree $\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{\delta+1}+2$. Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$. In particular we show that if $d^3 \log n \leq o(n)$ then every $d$-regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$. We also prove that any graph with $n$ vertices and minimum degree $\delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{\delta+1}+2$ times.