On the connectivity of bipartite distance-balanced graphs

Miklavič, Štefko; Šparl, Primož

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Title:	On the connectivity of bipartite distance-balanced graphs
Authors:	ID Miklavič, Štefko (Author) ID Šparl, Primož (Author)
Files:	http://dx.doi.org/10.1016/j.ejc.2011.10.002
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	A connected graph $\varGamma$ is said to be distance-balanced whenever for any pair of adjacent vertices $u,v$ of $\varGamma$ 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 $k \geq 3$ 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.
Keywords:	graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Year of publishing:	2012
Number of pages:	str. 237-247
Numbering:	Vol. 33, no. 2
PID:	20.500.12556/RUP-1686
ISSN:	0195-6698
UDC:	519.17
COBISS.SI-ID:	1024369748
Publication date in RUP:	15.10.2013
Views:	4723
Downloads:	98
Metadata:
:	MIKLAVIČ, Štefko and ŠPARL, Primož, 2012, On the connectivity of bipartite distance-balanced graphs. [online]. 2012. Vol. 33, no. 2, p. 237–247. [Accessed 19 March 2025]. Retrieved from: http://dx.doi.org/10.1016/j.ejc.2011.10.002 Copy citation