On the connectivity of bipartite distance-balanced graphs

Abstract

A connected graph Γ is said to be distance-balanced whenever for any pair of adjacent vertices u , v of Γ the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u . In [K. Handa, Bipartite graphs with balanced ( a , b ) -partitions, Ars Combin. 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 ≥ 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.