On the connectivity of bipartite distance-balanced graphs

Štefko Miklavič (Author), Primož Šparl (Author)

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

teorija grafov;povezani grafi;povzanost;razdaljno uravnoteženi grafi;dvodelni grafi;graph theory;connected graphs;connectivity;distance-balanced graphs;bipartite graphs;

Data

Language:	English
Year of publishing:	2012
Typology:	1.01 - Original Scientific Article
Organization:	UL PEF - Faculty of Education
UDC:	519.17
COBISS:	1024369748
ISSN:	0195-6698
Views:	849
Downloads:	256
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	O povezanosti dvodelnih razdaljno uravnoteženih grafov
Secondary keywords:	teorija grafov;povezani grafi;povzanost;razdaljno uravnoteženi grafi;dvodelni grafi;
Type (COBISS):	Not categorized
Pages:	str. 237-247
Volume:	ǂVol. ǂ33
Issue:	ǂno. ǂ2
Chronology:	2012
ID:	1477170