On the maximum local mean order of sub-k-trees of a k-tree

TL;DR

This paper investigates the maximum local mean order of sub-k-trees in k-trees, confirming that it occurs at a non-major k-clique, extending previous results from trees to k-trees.

Contribution

It proves that in any k-tree, the maximum local mean order of sub-k-trees containing a k-clique is achieved at a non-major k-clique, answering a recent open question.

Findings

Maximum local mean order occurs at a non-major k-clique.

Extends known results from trees to k-trees.

Provides an affirmative answer to a 2018 open problem.

Abstract

For a k-tree T, a generalization of a tree, the local mean order of sub-k-trees of T is the average order of sub-k-trees of T containing a given k-clique. The problem whether the largest local mean order of a tree (i.e., a 1-tree) at a vertex always takes on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. In 2018, Stephens and Oellermann asked a similar problem: for any k-tree T, does the maximum local mean order of sub-k-trees containing a given k-clique occur at a k-clique that is not a major k-clique of T? In this paper, we give it an affirmative answer.