| Title: | On the connectivity of bipartite distance-balanced graphs |
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| Authors: | ID Miklavič, Štefko (Author)
ID Šparl, Primož (Author) |
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| Files: |  http://dx.doi.org/10.1016/j.ejc.2011.10.002
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| Language: | English |
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| Work type: | Not categorized |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | IAM - Andrej Marušič Institute
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| 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. |
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| Keywords: | graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs |
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| Year of publishing: | 2012 |
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| Number of pages: | str. 237-247 |
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| Numbering: | Vol. 33, no. 2 |
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| PID: | 20.500.12556/RUP-1686  |
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| ISSN: | 0195-6698 |
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| UDC: | 519.17 |
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| COBISS.SI-ID: | 1024369748 |
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| Publication date in RUP: | 15.10.2013 |
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| Views: | 3964 |
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| Downloads: | 97 |
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