Crown Graph - Applications

Applications

In etiquette, a traditional rule for arranging guests at a dinner table is that men and women should alternate positions, and that no married couple should sit next to each other. The arrangements satisfying this rule, for a party consisting of n married couples, can be described as the Hamiltonian cycles of a crown graph. For instance, the arrangements of vertices shown in the figure can be interpreted as seating charts of this type in which each husband and wife are seated as far apart as possible. The problem of counting the number of possible seating arrangements, or almost equivalently the number of Hamiltonian cycles in a crown graph, is known in combinatorics as the ménage problem; for crown graphs with 6, 8, 10, ... vertices the number of (oriented) Hamiltonian cycles is

2, 12, 312, 9600, 416880, 23879520, 1749363840, ... (sequence A094047 in OEIS)

Crown graphs can be used to show that greedy coloring algorithms behave badly in the worst case: if the vertices of a crown graph are presented to the algorithm in the order u0, v0, u1, v1, etc., then a greedy coloring uses n colors, whereas the optimal number of colors is two. This construction is attributed to Johnson (1974); crown graphs are sometimes called Johnson’s graphs with notation Jn. Fürer (1995) uses crown graphs as part of a construction showing hardness of approximation of coloring problems.

Matoušek (1996) uses distances in crown graphs as an example of a metric space that is difficult to embed into a normed vector space.

As Miklavič & Poročnik (2003) show, crown graphs are one of a small number of different types of graphs that can occur as distance-regular circulant graphs.

Agarwal et al. (1994) describe polygons that have crown graphs as their visibility graphs; they use this example to show that representing visibility graphs as unions of complete bipartite graphs may not always be space-efficient.

A crown graph with 2n vertices, with its edges oriented from one side of the bipartition to the other, forms the standard example of a partially ordered set with order dimension n.