Notes on Spreads of Degrees in Graphs

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.