Abstract
We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We show that a 2-connected cubic graph G with oddness ω (G) different from the Petersen graph has order at least 5.41 ω (G), and for any integer k with 2 ≤ k ≤ 6 we construct an infinite family of cubic graphs with cyclic connectivity k and small oddness ratio ¦V(G)¦/ω(G). For cyclic connectivity 2, 4, 5, and 6 we improve the upper bounds on the oddness ratio of snarks to 7.5, 13, 25, and 99 from the known values 9, 15, 76, and 118, respectively. We also construct a cyclically 4-connected snark of girth 5 with oddness 4 and order 44, improving the best previous value of 46.
The authors acknowledge partial support from the APW grant ESF-EC-0009-10 within the EU-ROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation, from APW-0223-10, and from VEGA 1/1005/12.
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© 2013 Scuola Normale Superiore Pisa
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Lukot’ka, R., Máčajová, E., Mazák, J., Škoviera, M. (2013). Snarks with large oddness and small number of vertices. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_10
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DOI: https://doi.org/10.1007/978-88-7642-475-5_10
Publisher Name: Edizioni della Normale, Pisa
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