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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 λe(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 λe(G) ≤ \frac{m + (k-1)t}{2k-1} and that if k ≥ 2, then λe (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 λe}(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.

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

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