The asymptotic connectivity of labelled coloured regular bipartite graphs

Abstract

A labelled coloured bipartite graph, of LCBG, is a bipartite (simple) graph whose vertices have been 2-coloured and the vertices of each colour labelled independently. It is shown that for fixed r⩾3 the proportion of r-regular LCBGs on 2n vertices which are r-connected approaches 1 as n → ∞. Also, fix r⩾3 and q>0; let g=max(4,2{q/(2(r−2))}). Then the numbers of the following types of r-regular LCBGs with 2n vertices are asymptotically equal as n → ∞: those with girth at least g; those which are cyclically-q-edge-connected; and those which are cyclically-q-vertex-connected.