Crown graph

In graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u1, u2, ..., un} and {v1, v2, ..., vn} and with an edge from ui to vj whenever i ≠ j. In graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u1, u2, ..., un} and {v1, v2, ..., vn} and with an edge from ui to vj whenever i ≠ j. The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product Kn × K2, as the complement of the Cartesian direct product of Kn and K2, or as a bipartite Kneser graph Hn,1 representing the 1-item and (n − 1)-item subsets of an n-item set, with an edge between two subsets whenever one is contained in the other. The 6-vertex crown graph forms a cycle, and the 8-vertex crown graph is isomorphic to the graph of a cube.In the Schläfli double six, a configuration of 12 lines and 30 points in three-dimensional space, the twelve lines intersect each other in the pattern of a 12-vertex crown graph. The number of edges in a crown graph is the pronic number n(n − 1). Its achromatic number is n: one can find a complete coloring by choosing each pair {ui, vi} as one of the color classes. Crown graphs are symmetric and distance-transitive. Archdeacon et al. (2004) describe partitions of the edges of a crown graph into equal-length cycles. The 2n-vertex crown graph may be embedded into four-dimensional Euclidean space in such a way that all of its edges have unit length. However, this embedding may also place some non-adjacent vertices a unit distance apart. An embedding in which edges are at unit distance and non-edges are not at unit distance requires at least n − 2 dimensions. This example shows that a graph may require very different dimensions to be represented as a unit distance graphs and as a strict unit distance graph. The minimum number of complete bipartite subgraphs needed to cover the edges of a crown graph (its bipartite dimension, or the size of a minimum biclique cover) is the inverse function of the central binomial coefficient. The complement graph of a 2n-vertex crown graph is the Cartesian product of complete graphs K2 ◻ {displaystyle square } Kn, or equivalently the 2 × n rook's graph. 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