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

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 maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. 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.