Reducing the maximum degree of a graph: comparisons of bounds

TL;DR

This paper compares bounds on the minimum vertices or edges to remove from a graph to reduce its maximum degree, analyzing when each bound is tighter and exploring potential for similar bounds in other graph parameters.

Contribution

It introduces and compares bounds on vertex and edge removal for degree reduction, providing insights into their relative effectiveness and potential for generalization.

Findings

Bounds are tight for disjoint star graphs.

Comparison identifies cases where each bound is preferable.

Framework suggests similar bounds may exist for other parameters.

Abstract

Let $\lambda(G)$ be the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed from $G$ for the same purpose. Let $k$ be the maximum degree of $G$, let $t$ be the number of vertices of degree $k$, let $M(G)$ be the set of vertices of degree $k$, let $n$ be the number of vertices in the closed neighbourhood of $M(G)$, and let $m$ be the number of edges incident to vertices in $M(G)$. Fenech and the author showed that $\lambda(G) \leq \frac{n+(k-1)t}{2k}$, and they essentially showed that $\lambda (G) \leq n \left ( 1- \frac{k}{k+1} { \Big( \frac{n}{(k+1)t} \Big) }^{1/k} \right )$. They also showed that $\lambda_{\rm e}(G) \leq \frac{m + (k-1)t}{2k-1}$ and $\lambda_{\rm e} (G) \leq m \left ( 1- \frac{k-1}{k} { \Big( \frac{m}{kt} \Big) }^{1/(k-1)} \right )$. These bounds are attained if $k \geq 2$ and $G$ is the union of $t$ pairwise vertex-disjoint $(k+1)$-vertex stars. For each of $\lambda(G)$ and $\lambda_{\rm e}(G)$, the two bounds on the parameter are compared for the purpose of determining, for each bound, the cases in which the bound is better than the other. This work is also motivated by the likelihood that similar pairs of bounds will be discovered for other graph parameters and the same analysis can be applied.