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On minimal graphs of diameter 2 with every edge in a 3-cycle. (English) Zbl 0603.05040
V. Chvátal and C. Thomassen [Distances in orientations of graphs, J. Comb. Theory, Ser. B 24, 61-75 (1978; Zbl 0311.05115)] proved that every bridgeless graph G of diameter 2 admits an orientation of diameter at most 6. The only difficult part of the proof is in the case that G has a minimal spanning bridgeless subgraph G’ with diameter 2 and with every edge in a 3-cycle. (Minimal here means that the diameter of G’-e is larger than 2 for all edges e.) A few years ago the author conjectured there are no graphs such as G’; if true the above proof would be greatly simplified. However, the conjecture is false and the main purpose of this article is to present an infinite class of counterexamples. He also points out that there are infinitely many planar graphs of this type of diameter k for each $$k\geq 3$$.
Reviewer: R.L.Hemminger

##### MSC:
 05C99 Graph theory 05C35 Extremal problems in graph theory
##### Keywords:
3-cycleless; diameter; counterexamples
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##### References:
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