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Shortness coefficient of cyclically 4-edge-connected cubic graphs. (English) Zbl 1432.05059
Summary: B. Grünbaum and J. Malkevitch [Aequationes Math. 14, 191–196 (1976; Zbl 0331.05118)] proved that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most \(\frac{76}{77}\). Recently, this was improved to \(\frac{359}{366}\) \((<\frac{52}{53})\) and the question was raised whether this can be strengthened to \(\frac{41}{42}\), a natural bound inferred from one of the Faulkner-Younger graphs. We prove that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most \(\frac{37}{38}\) and that we also get the same value for cyclically 4-edge-connected cubic graphs of genus \(g\) for any prescribed genus \(g \geqslant 0\). We also show that \(\frac{45}{46}\) is an upper bound for the shortness coefficient of cyclically 4-edge-connected cubic graphs of genus \(g\) with face lengths bounded above by some constant larger than 22 for any prescribed \(g \geqslant 0\).
MSC:
05C40 Connectivity
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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