تقرير
Decreasing the mean subtree order by adding $k$ edges
| العنوان: | Decreasing the mean subtree order by adding $k$ edges |
|---|---|
| المؤلفون: | Cambie, Stijn, Chen, Guantao, Hao, Yanli, Tokar, Nizamettin |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, 05C05, 05C35, 05C40 |
| الوصف: | The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\mu(H) > \mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\mu(H) < \mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\mu(H) < \mu(G)$. Furthermore, they proposed that $\mu(K_m+nK_1) < \mu(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures. Comment: 11 Pages, 5 Figures Paper identical to JGT submission |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2308.12808 |
| رقم الأكسشن: | edsarx.2308.12808 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |