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Special Classes of Snarks

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Abstract

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.

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References

  1. Behzad, M.: Graphs and their chromatic numbers, PhD Thesis, Michigan State Univ., 1965.

  2. Brinkmann, G. and Steffen, E.: 3-and 4-critical graphs of small even order, Discrete Math. 169 (1997), 193–197.

    Google Scholar 

  3. Brinkmann, G. and Steffen, E.: Snarks and reducibility, Ars Combin. 50 (1998), 292–296.

    Google Scholar 

  4. Brinkmann, G. and Steffen, E.: Chromatic-index-critical graphs of orders 11 and 12, European J. Combin. 19(8) (1998), 889–900.

    Google Scholar 

  5. Cai, L.: A snark of order 24, Preprint.

  6. Cavicchioli, A., Meschiari, M., Ruini, B. and Spaggiari, F.: A survey on snarks and new results: products, reducibility and a computer search, J. Graph Theory 28(2) (1998), 57–86.

    Google Scholar 

  7. Celmins, U. A. and Swart, E. R.: The construction of snarks, Dept. of Combinatorics and Optimization, Univ. of Waterloo, Research Report CORR (1979), 79-18.

  8. Chetwynd, A. G. and Wilson, R. J.: Snarks and supersnarks, In: G. Chartrand (ed.), Theory and Applications of Graphs, Wiley-Interscience, New York, 1981, pp. 215–241.

    Google Scholar 

  9. Collier, J. B. and Schmeichel, E. F.: Systematic searches for hypohamiltonian graphs, Networks 8 (1978), 193–200.

    Google Scholar 

  10. Fiorini, S.: Hypohamiltonian snarks, In: M. Fiedler (ed.), Graphs and Other Combinatorial Topics, Proc. Third Czechoslovak Sympos. on Graph Theory, Prague, 1982, Teubner-Texte zur Math. 59, Teubner, Leipzig, 1983, pp. 70–75.

    Google Scholar 

  11. Fiorini, S. and Wilson, R. J.: Edge-Colourings of Graphs, Res. Notes in Math. 16, Pitman, London, 1977.

    Google Scholar 

  12. Fouquet, J. L.: Note sur la non existence d'un snark d'ordre 16, Discrete Math. 38 (1982), 163–171.

    Google Scholar 

  13. Grünewald, S. and Steffen, E.: Cyclically 5-edge connected non-bicritical critical snarks, Discussiones Math.-Graph Theory 19(1) (1999), 5–11.

    Google Scholar 

  14. Grünewald, S. and Steffen, E.: Chromatic-index-critical graphs of even order, J. Graph Theory 30(1) (1999), 27–36.

    Google Scholar 

  15. Häggkvist, R. and Chetwynd, A.: Some upper bounds on the total and list chromatic numbers of multigraphs, J. Graph Theory 16 (1992), 503–516.

    Google Scholar 

  16. Hind, H.: An improved bound for the total chromatic number of a graph, Graphs Combin. 6 (1990), 153–159.

    Google Scholar 

  17. Hind, H. and Molloy, M.: Total coloring with ( +poly log colors, SIAM J. Comput. 28(3) (1999), 816–821.

    Google Scholar 

  18. Isaacs, R.: Infinite families of nontrivial trivalent graphs which are not Tait colorable, Amer. Math. Monthly 82 (1975), 221–239.

    Google Scholar 

  19. Jaeger, F.: Nowhere-zero flow problem, In: L. W. Beineke and R. J. Wilson (eds), Selected Topics in Graph Theory, Academic Press, London, 1988, pp. 71–95.

    Google Scholar 

  20. Jensen, J. R. and Toft, B.: Graph Coloring Problems, Wiley, New York, 1995.

    Google Scholar 

  21. Kochol, M.: Snarks without small cycles, J. Combin. Theory B 67 (1996), 34–47.

    Google Scholar 

  22. Möller, M., Carstens, H. G. and Brinkmann, G.: Nowhere-zero flows in low genus graphs, J. Graph Theory 12 (1988), 183–190.

    Google Scholar 

  23. Molloy, M. and Reed, B.: A bound on the total chromatic number, Combinatorica 18(2) (1998), 241–280.

    Google Scholar 

  24. Nedela, R. and Škoviera, M.: Decompositions and reductions of snarks, J. Graph Theory 22(3) (1996), 253–279.

    Google Scholar 

  25. Petersen, J.: Sur le théoréme de Tait, L'intermédiaire des Mathématicien 5 (1898), 225–227.

    Google Scholar 

  26. Preissmann, M.: Snarks of order 18, Discrete Math. 42 (1982), 125–126.

    Google Scholar 

  27. Sanders, P. D. and Yue, Zhao: On total 9-coloring planar graphs of maximum degree seven, J. Graph Theory 31(1) (1999), 67–73.

    Google Scholar 

  28. Seymour, P. D.: Nowhere-zero 6-flows, J. Combin. Theory B 30 (1981), 130–135.

    Google Scholar 

  29. Sinclair, P. A.: The construction and reduction of strong snarks, Discrete Math. 167 (1997), 553–570.

    Google Scholar 

  30. Steffen, E.: Counterexamples to a conjecture about bottlenecks in non-Tait-colourable cubic graphs, Discrete Math. 161 (1996), 315.

    Google Scholar 

  31. Steffen, E.: Tutte's 5-Flow conjecture for graphs of nonorientable genus 5, J. Graph Theory 224)(1996), 309–319.

    Google Scholar 

  32. Steffen, E.: Classifications and characterizations of snarks, Discrete Math. 188 (1998), 183–203

    Google Scholar 

  33. Steffen, E.: On bicritical snarks, Math. Slovaca 51 (2001), 141–150.

    Google Scholar 

  34. Steffen, E.: Non-bicritical critical snarks, Preprint 97-042, SFB 343, Diskrete Strukturen in der Mathematik, Universität Bielefeld, 1997.

  35. Vizing, V.: Some unsolved problems in graph theory, Russian Math. Surveys 23 (1968), 125–141.

    Google Scholar 

  36. Zhang, Z. F. and Wang, J. F.: The progress of total colorings of graphs, Adv. in Math. 4 (1992), 390–397.

    Google Scholar 

  37. Watkins, J. J. and Wilson, R. J.: A survey of snarks, In: Graph Theory, Combinatorics, and Applications (Kalamazoo, MI, 1988), Wiley-Interscience, New York, 1991, pp. 1129–1144.

    Google Scholar 

  38. Xu, B.: A sufficient condition for bipartite graphs to be of type one, J. Graph Theory 29(3) (1998), 133–137.

    Google Scholar 

  39. Yap, H. P.: Total Colourings of Graphs, Lecture Notes in Math. 1623, Springer, New York, 1996.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100, Modena, Italy

    A. Cavicchioli

  2. Dipartimento di Matematica Pura ed Applicata, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100, Modena, Italy

    T. E. Murgolo, B. Ruini & F. Spaggiari

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  1. A. Cavicchioli
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  2. T. E. Murgolo
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  3. B. Ruini
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  4. F. Spaggiari
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Cavicchioli, A., Murgolo, T.E., Ruini, B. et al. Special Classes of Snarks. Acta Applicandae Mathematicae 76, 57–88 (2003). https://doi.org/10.1023/A:1022864000162

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