20.500.12556/RUP-1686 On the connectivity of bipartite distance-balanced graphs 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. graph theory connected graphs connectivity distance-balanced graphs bipartite graphs teorija grafov povezani grafi povzanost razdaljno uravnoteženi grafi dvodelni grafi true true false Angleški jezik Slovenski jezik Delo ni kategorizirano 2013-10-15 12:06:17 2013-10-15 12:06:17 2024-03-01 12:01:40 0000-00-00 00:00:00 2012 0 0 str. 237-247 no. 2 Vol. 33 2012 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 0195-6698 519.17 1024369748 http://dx.doi.org/10.1016/j.ejc.2011.10.002 1 https://repozitorij.upr.si/Dokument.php?lang=slv&id=1686 Inštitut Andrej Marušič 0 0 0