Bounding mean orders of sub-$k$-trees of $k$-trees

TL;DR

This paper establishes bounds on the mean order of sub-$k$-trees in $k$-trees, identifying extremal structures and solving open problems related to their mean order.

Contribution

It provides new bounds and characterizations for the mean order of sub-$k$-trees, resolving open questions in the study of $k$-trees.

Findings

Maximum local mean order attained in a degree 1 $k$-clique

Global mean order is at most twice the maximum

Large $k$-stars minimize the global mean order

Abstract

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.