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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 \(g \geqslant 0\).
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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