Uniformly connected graphs

TL;DR

This paper explores the relationship between uniformly $k$-connected and uniformly $k$-edge-connected graphs, establishing new results for their equivalence, structure, and bounds on vertices for specific values of $k$.

Contribution

It provides a constructive characterization of uniformly 3-connected graphs inspired by Tutte's Wheel Theorem and clarifies the relationship between these classes for different $k$ values.

Findings

Uniformly $k$-connected graphs are also uniformly $k$-edge-connected for $k\, extless= 3$.

The paper demonstrates that the equivalence does not hold for $k>3$.

A tight bound on the number of vertices in minimum degree for uniformly 3-connected graphs is established.

Abstract

In this article we investigate the structure of uniformly $k$-connected and uniformly $k$-edge-connected graphs. Whereas both types have previously been studied independent of each other, we analyze relations between these two classes. We prove that any uniformly $k$-connected graph is also uniformly $k$-edge-connected for $k\le 3$ and demonstrate that this is not the case for $k>3$. Furthermore, uniformly $k$-connected and uniformly $k$-edge-connected graphs are well understood for $k\le 2$ and it is known how to construct uniformly $3$-edge-connected graphs. We contribute here a constructive characterization of uniformly $3$-connected graphs that is inspired by Tuttes Wheel Theorem. Eventually, these results help us to prove a tight bound on the number of vertices of minimum degree in uniformly $3$-connected graphs.