Home > Journals > TAG > Vol. 8 > Iss. 1 (2021)

## Publication Date

April 2021

## Abstract

Let λ(*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 λ_{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 that have at least one vertex in M(G). Fenech and the author showed that λ(*G*) ≤ *{n+(k-1)t}{2k}*, and they essentially showed that λ (*G*) ≤ n ( 1- \frac{k}{k+1} { \Big( \frac{n}{(k+1)t} \Big) }^{1/k} \right ). They also showed that λ_{e}(*G*) ≤ \frac{m + (k-1)t}{2k-1} and that if *k ≥ 2*, then λ_{e} (*G*) ≤ m ( 1- \frac{k-1}{k} { \Big( \frac{m}{kt} \Big) }^{1/(k-1)} \right ). These bounds are attained if *G* is the union of pairwise vertex-disjoint (*k*+1)-vertex stars. In this paper, we determine the cases in which one bound on λ(G) is better than the other, and we show that the first bound on λ_{e}}(*G*) is better than the second. This work is motivated by the likelihood that similar pairs of bounds will be discovered for other graph parameters and the same analysis can be applied.

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

## Recommended Citation

Borg, Peter
(2021)
"Reducing the maximum degree of a graph: comparisons of bounds,"
*Theory and Applications of Graphs*: Vol. 8:
Iss.
1, Article 6.

DOI: 10.20429/tag.2021.080106

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol8/iss1/6