Wide diameter of Cartesian graph bundles

Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width ▫$k$▫ of a graph ▫$G$▫, ▫$k$▫-diameter, is defined as the minimum integer ▫$d$▫ for which there exist at least ▫$k$▫ internally disjoint paths of length at most ▫$d$▫ between any two distinct vertices in ▫$G$▫. Denote by ▫${mathscr D}_c(G)$▫ the ▫$c$▫-diameter of ▫$G$▫ and ▫$kappa(G)$▫ the connectivity of ▫$G$▫. In the context of computer networks, wide diameters of Cartesian graph products have been recently studied by many authors. Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let ▫$G$▫ be a Cartesian graph bundle with fiber ▫$F$▫ over base ▫$B$▫, ▫$0 < a le kappa(F)$▫, and ▫$0 < b le kappa(B)$▫. We prove that ▫${mathscr D}_{a+b}(G) le {mathscr D}_a(F) + {mathscr D}_b(B) + 1$▫. Moreover, if ▫$G$▫ is a graph bundle with fiber ▫$F ne K_2$▫ over base ▫$B ne K_2$▫, then ▫${mathscr D}_{a+b}(G) le {mathscr D}_a(F) + {mathscr D}_b(B)$▫. The bounds are tight