Counting Connected Partitions of Graphs

TL;DR

This paper investigates the number of ways to partition a connected graph into connected subgraphs with specified edge counts, providing tight lower bounds based on graph parameters and exploring related vertex partition counts.

Contribution

It introduces new lower bounds on the number of connected partitions of graphs, extending the understanding of graph decompositions with tight bounds related to graph degree and cuts.

Findings

Lower bounds on $P(G,k)$ as a function of $n$, $d$, and $ ext{CMC}_r(G)$

Tight bounds up to a constant factor for these lower bounds

The number of vertex partitions into connected parts is $ ext{Omega}(d^{k-1})$

Abstract

Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$ such that every $m_i$ is realized by a connected subgraph $H_i$ of $G$ with $m_i$ edges such that $\cup_{i=1}^kE(H_i)=E(G)$. We also consider the vertex-partition analogue. We prove various lower bounds on $P(G,k)$ as a function of the number $n$ of vertices in $G$, as a function of the average degree $d$ of $G$, and also as the size $\mathrm{CMC}_r(G)$ of $r$-partite connected maximum cuts of $G$. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number $\pi(G,k)$ of unordered $k$-tuples with $\sum_{i=1}^kn_i=n$, that are realizable by vertex partitions into $k$ connected parts of respective sizes $n_1,n_2,\dots,n_k$, is $\Omega(d^{k-1})$.