On the Connectivity of Bipartite Distance-Balanced Graphs

TL;DR

This paper investigates the structure of bipartite distance-balanced graphs, disproves a conjecture about their connectivity, and classifies certain non-3-connected cases, revealing their regularity and uniqueness.

Contribution

It answers Handa's question negatively by providing counterexamples, classifies minimal non-3-connected bipartite distance-balanced graphs, and characterizes their structural properties.

Findings

Counterexamples to Handa's conjecture exist.

Minimal non-3-connected bipartite distance-balanced graphs have 18 vertices and are unique.

All such graphs with a certain property are regular and form infinite families.

Abstract

A connected graph $\G$ is said to be {\it distance-balanced} whenever for any pair of adjacent vertices $u,v$ of $\G$ the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$. In [Bipartite graphs with balanced $(a,b)$-partitions, {\em Ars Combin.} {\bf 51} (1999), 113-119] Handa asked whether every bipartite distance-balanced graph, that is not a cycle, is 3-connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distance-balanced graph, that is not a cycle and is not 3-connected, has 18 vertices and is unique. In addition, we give a complete classification of non-3-connected bipartite distance-balanced graphs for which the minimal distance between two vertices in a 2-cut is three. All such graphs are regular and for each $k \geq 3$ there exists an infinite family of such graphs which are $k$-regular. Furthermore, we determine a number of structural properties that a bipartite distance-balanced graph, which is not 3-connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distance-balanced graphs.