Crossing Graphs as Joins of Graphs and Cartesian Products of Median Graphs

Abstract

For a partial cube $G$ its crossing graph $G^\#$ is the graph whose vertices are the T-classes of $G$, two classes being adjacent if they cross on some cycle in $G$. The following problem posed in [S. Klavzcar and H. M. Mulder, SIAM J. Discrete Math., 15 (2002), pp. 235-251, Problem 7.1] is considered: What can be said about the partial cube $G$ if $G^\#$ is the join $A\oplus B$ of graphs $A$ and $B$ with at least one edge? It is proved that for arbitrary graphs $A$ and $B$, where at least one of them contains an edge, there exists a Cartesian prime partial cube $G$ such that $G^\# = A\oplus B$. On the other hand, if $G$ is a median graph, then $G^\# = A\oplus B$ if and only if $G=H\,\square\, K$, where $H^\# = A$ and $K^\# = B$. Along the way some new facts about partial cubes are obtained; for instance, a bipartite graph of radius 2 is a partial cube if and only if it is $K_{2,3}$-free.