In this paper we show that for a given $k$-tree $T$ with a $k$-clique $C$, the local mean order of all sub-$k$-trees of $T$ containing $C$ is not less than the global mean order of all sub-$k$-trees of $T$, and the path-type $k$-trees have the smallest global mean sub-$k$-tree order among all $k$-trees of a given order. These two results give solutions to two problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees. Furthermore, the mean sub-$k$-tree order as a function on $k$-trees is shown to be monotone with respect to inclusion. This generalizes Jamison's result for the case $k=1$ [J. Combin. Theory Ser. B 35 (1983), 207-223].
@article{10_37236_11280,
author = {Zuwen Luo and Kexiang Xu},
title = {On the local and global mean orders of sub-\(k\)-trees of \(k\)-trees},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11280},
zbl = {1518.05041},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11280/}
}
TY - JOUR
AU - Zuwen Luo
AU - Kexiang Xu
TI - On the local and global mean orders of sub-\(k\)-trees of \(k\)-trees
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11280/
DO - 10.37236/11280
ID - 10_37236_11280
ER -
%0 Journal Article
%A Zuwen Luo
%A Kexiang Xu
%T On the local and global mean orders of sub-\(k\)-trees of \(k\)-trees
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11280/
%R 10.37236/11280
%F 10_37236_11280
Zuwen Luo; Kexiang Xu. On the local and global mean orders of sub-\(k\)-trees of \(k\)-trees. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11280
