Odd-4 Graph, Fano Planes, and the Coxeter Graph

The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the odd-2 graph is the pentagon.
A Fano plane is a collection of seven of these triplets where no two triplets share two numbers. The Fano plane is a structurally unique 7 point configuration, with 30 distinct numberings when the numbers 1-7 are assigned to the seven points. If a Fano plane is removed from the odd-4 graph, the result is the Coxeter graph, a non-Hamiltonian cubic symmetric graph with many interesting properties. If connections to the fifteen Fano planes of a single color are added instead, the result is the Hoffman–Singleton graph, which is also called the (7, 5)-cage.


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Late in his career, Coxeter saw this graph in a technical paper, and found several interesting properties about it. He asked his assistant Asia what it was called. "You discovered it, 30 years ago. It's popularly called the Coxeter graph." Coxeter then wrote up a paper, "My Graph".
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