On the connectivity of bipartite distance-balanced graphs

Miklavič, Štefko (author), Šparl, Primož (author)

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.

• Language: English
• Keywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
• Work type: Not categorized
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education
• Year: 2012
• Number of pages: Str. 237-247
• Numbering: Vol. 33, no. 2
• PID: 20.500.12556/RUL-84784
• UDC: 519.17
• ISSN on article: 0195-6698
• COBISS.SI-ID: 1024369748
• Publication date in RUL: 09.09.2016

Record is a part of a journal

• Title: European journal of combinatorics
• Shortened title: Eur. j. comb.
• Publisher: Academic Press
• ISSN: 0195-6698
• COBISS.SI-ID: 25427968

Secondary language

• Language: Slovenian
• Title: O povezanosti dvodelnih razdaljno uravnoteženih grafov
• Keywords: teorija grafov, povezani grafi, povzanost, razdaljno uravnoteženi grafi, dvodelni grafi