Cubic graph

A graph whose every vertex has degree of a vertex three.

The bipartite cubic graph with vertices $A_i,B_i$, $i\in \mathbb{Z}/7\mathbb{Z}$, and edges $A_i{B}_i,A_i{B}_{i+1},A_i{B}_{i-2}$. The three offsets give a Tait coloring.

Triangulate a torus using the seven vertices of the complete graph $K_7$ and oriented triangles $(i,i+1,i+3)$ and $(i,i+3,i+2)$, with indices modulo seven. Every edge occurs twice with opposite orientations; each vertex has a cyclic link. The Euler characteristic is $7-21+14=0$. 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.