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Flower Snark


FlowerSnarks

The flower snarks, denoted J_n for n=5, 7, 9, ..., are a family of graphs discovered by Isaacs (1975) which are snarks. They are precisely the twisted Isaacs graphs FS(2,n) for odd n>=5, whose closing permutation is a transposition (Fouquet et al. 2010).

J_5 appears in Scheinerman and Ullman (2011, p. 96) as an example of a graph with edge chromatic number and fractional edge chromatic number (4 and 3, respectively) both integers but not equal.

Flower snarks are unit-distance. They are also maximally nonhamiltonian (Clark and Entringer 1983) as well as platypus graphs.

The graph crossing number of J_5 is 4, while that of J_n is n for every odd n>=7 (Zheng et al. 2008, Lemmas 4.3 and 4.8).

Precomputed properties of flower snarks are implemented in the Wolfram Language as GraphData[{"Flower", n}].


See also

Flower Graph, Snark, Twisted Isaacs Graph

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References

Bondy, J. A. and Murty, U. S. R. Graph Theory. Berlin, Germany: Springer-Verlag, p. 462, 2008.Clark, L. and Entringer, R. "Smallest Maximally Nonhamiltonian Graphs." Periodica Math. Hungarica 14, 57-68, 1983.Fouquet, J.-L.; Thuillier, H.; and Vanherpe, J.-M. "On a Family of Cubic Graphs Containing the Flower Snarks." Discuss. Math. Graph Theory 30, 289-314, 2010. https://doi.org/10.7151/dmgt.1495.Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, pp. 82 and 97-100, 1993.House of Graphs. Flower Snarks. Flower Snark J5, Flower Snark J7, Flower Snark J9, and Flower Snark J11.Isaacs, R. "Infinite Families of Nontrivial Trivalent Graphs Which Are Not Tait Colorable." Amer. Math. Monthly 82, 221-239, 1975.Scheinerman, E. R. and Ullman, D. H. Fractional Graph Theory A Rational Approach to the Theory of Graphs. New York: Dover, 2011.West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 306, 2000.Yang, X. "The Crossing Number of Flower Snark and K_m^- square P_n." Master's thesis. Dalian, China: Dalian University of Technology, 2007. In Chinese.Zheng, W.; Lin, X.; Yang, Y.; and Yang, X. "The Crossing Number of Flower Snarks and Related Graphs." Ars Combin. 86, 57-64, 2008. https://combinatorialpress.com/ars-articles/volume-086-ars-articles/the-crossing-number-of-flower-snarks-and-related-graphs/.

Referenced on Wolfram|Alpha

Flower Snark

Cite this as:

Weisstein, Eric W. "Flower Snark." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FlowerSnark.html

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