Concept

Heawood graph

In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood. The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6. It is a distance-transitive graph (see the Foster census) and therefore distance regular. There are 24 perfect matchings in the Heawood graph; for each matching, the set of edges not in the matching forms a Hamiltonian cycle. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, 3-color its edges) in eight different ways. Every two perfect matchings, and every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph. There are 28 six-vertex cycles in the Heawood graph. Each 6-cycle is disjoint from exactly three other 6-cycles; among these three 6-cycles, each one is the symmetric difference of the other two. The graph with one node per 6-cycle, and one edge for each disjoint pair of 6-cycles, is the Coxeter graph. The Heawood graph is a toroidal graph; that is, it can be embedded without crossings onto a torus. One embedding of this type places its vertices and edges into three-dimensional Euclidean space as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus, the Szilassi polyhedron. Another embedding is the following: Tile the plane using regular hexagons, and color them periodically with 7 colors. For any color, the hexagons of this color form a triangular grid. In this grid, one can take a parallelogram with centres in the middle of the hexagons of the chosen color.

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Related concepts (15)
Coxeter graph
In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter. The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth 7. It is also a 3-vertex-connected graph and a 3-edge-connected graph. It has book thickness 3 and queue number 2. The Coxeter graph is hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian.
Symmetric graph
In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u_1—v_1 and u_2—v_2 of G, there is an automorphism such that and In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1-arc-transitive or flag-transitive. By definition (ignoring u_1 and u_2), a symmetric graph without isolated vertices must also be vertex-transitive.
Distance-transitive graph
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith. A distance-transitive graph is interesting partly because it has a large automorphism group.
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