Given two positive integers and , the bipartite Kneser graph is the graph whose two bipartite sets of vertices represent the -subsets and -subsets of and where two vertices are connected if and only if they are in different sets and one is a subset of the other. therefore has vertices and is regular of degree .
By definition, is isomorphic to .
is the bipartite double graph of the Kneser graph .
is connected for . Simpson (1991) showed that the bipartite Kneser graphs are symmetric. Shields and Savage showed that is Hamiltonian for .
is isomorphic to the -crown graph, is isomorphic to the order -ladder rung graph, and is isomorphic to the bipartite double graph of the odd graph . Graphs of the latter class are distance-transitive, and therefore also distance-regular (Brouwer et al. 1989, p. 222).
Special cases are summarized in the following table.
graph | |
cycle graph | |
cubical graph | |
Desargues graph | |
Danzer graph | |
crown graph | |
bipartite double graph of the odd graph | |
-ladder rung graph | |
middle layer graph |
It has long been conjectured that is Hamiltonian for , and this was verified by Shields and Savage for .
The -bipartite Kneser graph is implemented in the Wolfram Language as GraphData["BipartiteKneser", n, k].