Component order edge connectivity, vertex degrees, and integer partitions

TL;DR

This paper investigates the conditions on vertex degrees that imply lower bounds on the k-component order edge connectivity of graphs, linking graph connectivity with integer partitions and analyzing the complexity of these bounds.

Contribution

It introduces degree-based conditions to estimate lower bounds on k-component order edge connectivity and explores the complexity for bounds of 1, 2, and 3 or more, also relating to integer partitions.

Findings

Derived degree conditions for lower bounds of 1 and 2

Analyzed complexity increase for bounds of 3 or more

Proved new results on integer partitions related to graph connectivity

Abstract

Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the $k$-component order edge connectivity of a graph is NP-hard. We determine conditions on the vertex degrees of $G$ that can be used to imply a lower bound on the $k$-component order edge connectivity of $G$. We will discuss the process for generating such conditions for a lower bound of 1 or 2, and we explore how the complexity increases when the desired lower bound is 3 or more. In the process, we prove some related results about integer partitions.