Abstract
An edge e of a graph G is called deletable for some orientation o if the restriction of o to \(G-e\) is a strong orientation. In 2021, Hörsch and Szigeti proposed a new parameter for 3-edge-connected graphs, called the Frank number, which refines k-edge-connectivity. The Frank number is defined as the minimum number of orientations of G for which every edge of G is deletable in at least one of them. They showed that every 3-edge-connected graph has Frank number at most 7 and that in case these graphs are also 3-edge-colourable graphs the parameter is at most 3. Here we strengthen the latter result by showing that such graphs have Frank number 2, which also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number 2 and use them in an algorithm to computationally show that the Petersen graph is the only cyclically 4-edge-connected cubic graph up to 36 vertices having Frank number greater than 2.
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References
Barát, J., Blázsik, Z.: Quest for graphs of Frank number \(3\) (2022). https://doi.org/10.48550/arXiv.2209.08804
Brinkmann, G., Goedgebeur, J.: Generation of cubic graphs and snarks with large girth. J. Graph Theory 86(2), 255–272 (2017). https://doi.org/10.1002/jgt.22125
Brinkmann, G., Goedgebeur, J., Hägglund, J., Markström, K.: Generation and properties of snarks. J. Comb. Theory. Ser. B 103(4), 468–488 (2013). https://doi.org/10.1016/j.jctb.2013.05.001
Coolsaet, K., D’hondt, S., Goedgebeur, J.: House of graphs 2.0: a database of interesting graphs and more. Discret. Appl. Math. 325, 97–107 (2023). https://doi.org/10.1016/j.dam.2022.10.013
Goedgebeur, J., Máčajová, E., Renders, J.: Frank-Number (2023). https://github.com/JarneRenders/Frank-Number
Goedgebeur, J., Máčajová, E., Renders, J.: Frank number and nowhere-zero flows on graphs (2023). arXiv:2305.02133 [math.CO]
Goedgebeur, J., Máčajová, E., Škoviera, M.: Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44. ARS Math. Contemp. 16(2), 277–298 (2019). https://doi.org/10.26493/1855-3974.1601.e75
Goedgebeur, J., Máčajová, E., Škoviera, M.: The smallest nontrivial snarks of oddness 4. Discret. Appl. Math. 277, 139–162 (2020). https://doi.org/10.1016/j.dam.2019.09.020
Hörsch, F., Szigeti, Z.: Connectivity of orientations of 3-edge-connected graphs. Eur. J. Comb. 94, 103292 (2021). https://doi.org/10.1016/j.ejc.2020.103292
Jaeger, F.: A survey of the cycle double cover conjecture. In: Alspach, B.R., Godsil, C.D. (eds.) North-Holland mathematics studies, annals of discrete mathematics (27): cycles in graphs, vol. 115, pp. 1–12. North-Holland (1985). https://doi.org/10.1016/S0304-0208(08)72993-1
Nash-Williams, C.S.J.A.: On orientations, connectivity and odd-vertex-pairings in finite graphs. Canad. J. Math. 12, 555–567 (1960). https://doi.org/10.4153/CJM-1960-049-6. publisher: Cambridge University Press
Seymour, P.D.: On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte. Proc. London Math. Soc. 3(3), 423–460 (1979). https://doi.org/10.1112/plms/s3-38.3.423
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Goedgebeur, J., Máčajová, E., Renders, J. (2023). On the Frank Number and Nowhere-Zero Flows on Graphs. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_26
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