Triangulate a torus using the seven vertices of the complete graph and oriented triangles and , with indices modulo seven. Every edge occurs twice with opposite orientations; each vertex has a cyclic link. The Euler characteristic is . Its dual graph is the Heawood graph. This dual map has seven disc faces, each adjacent to all the others, so it needs seven face colors although its edges have a Tait coloring.
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The Heawood graph is a specific type of graph in graph theory that serves as an important example in various areas, including topology and combinatorics. It is named after the mathematician Percy John Heawood, who studied it in the context of map coloring problems. Here are some key features of the Heawood graph: 1. **Structure**: The Heawood graph is a bipartite graph with 14 vertices and 21 edges.