Notes on Spreads of Degrees in Graphs

TL;DR

This paper explores the spread of degrees in graphs, generalizing a fundamental result to larger sets and providing bounds based on graph properties, with applications to trees and outerplanar graphs.

Contribution

It proves a generalized lower bound for degree spreads in graphs, offers elementary proofs, and establishes sharp bounds for specific graph classes.

Findings

Proves that $sp(G,k) \,\geq\, k+2$ for graphs with at least $k+2$ vertices.

Provides lower bounds on $sp(G,k)$ based on degree measures.

Shows bounds are sharp for trees and maximal outerplanar graphs.

Abstract

Perhaps the very first elementary exercise one encounters in graph theory is the result that any graph on at least two vertices must have at least two vertices with the same degree. There are various ways in which this result can be non-trivially generalised. For example, one can interpret this result as saying that in any graph $G$ on at least two vertices there is a set $B$ of at least two vertices such that the difference between the largest and the smallest degrees (in $G$) of the vertices of $B$ is zero. In this vein we make the following definition. For any $B\subset V(G)$, let the spread $sp(B)$ of $B$ be defined to be the difference between the largest and the smallest of the degrees of the vertices in $B$. For any $k\geq 0$, let $sp(G,k)$ be the largest cardinality of a set of vertices $B$ such that $sp(B)\leq k$. Therefore the first elementary result in graph theory says that, for any graph $G$ on at least two vertices, $sp(G,0)\geq 2$. In this paper we first give a proof of a result of Erd\" os, Chen, Rousseau and Schelp which generalises the above to $sp(G,k)\geq k+2$ for any graph on at least $k+2$ vertices. Our proof is short and elementary and does not use the famous Erd\" os-Gallai Theorem on vertex degrees. We then develop lower bounds for $sp(G,k)$ in terms of the order of $G$ and its minimum, maximum and average degree. We then use these results to give lower bounds on $sp(G,k)$ for trees and maximal outerplanar graphs, most of which we show to be sharp.