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Structural and computational results on platypus graphs. (English) Zbl 1462.05110

Summary: A platypus graph is a non-hamiltonian graph for which every vertex-deleted subgraph is traceable. They are closely related to families of graphs satisfying interesting conditions regarding longest paths and longest cycles, for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian graphs.
In this paper, we first investigate cubic platypus graphs, covering all orders for which such graphs exist: in the general and polyhedral case as well as for snarks. We then present (not necessarily cubic) platypus graphs of girth up to 16 – whereas no hypohamiltonian graphs of girth greater than 7 are known – and study their maximum degree, generalising two theorems of G. Chartrand et al. [in: Second international conference on combinatorial mathematics, New York, 1978, Ann. New York Acad. Sci. 319, 130–135 (1979; Zbl 0481.05039)]. Using computational methods, we determine the complete list of all non-isomorphic platypus graphs for various orders and girths. Finally, we address two questions raised by the third author in [J. Graph Theory 86, No. 2, 223–243 (2017; Zbl 1370.05115)].

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C85 Graph algorithms (graph-theoretic aspects)
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References:

[1] Alspach, B. R., The classification of hamiltonian generalized Petersen graphs, J. Combin. Theory, Ser. B, 34, 293-312 (1983) · Zbl 0516.05034
[2] Boben, M.; Pisanski, T.; Žirnki, A., I-graphs and the corresponding configurations, J. Combin. Des., 13, 406-424 (2005) · Zbl 1076.05065
[3] Bondy, J. A., Variations on the hamiltonian theme, Canad. Math. Bull., 15, 57-62 (1972) · Zbl 0238.05115
[4] Available at http://hog.grinvin.org/ · Zbl 1292.05254
[5] Brinkmann, G.; Goedgebeur, J.; Hägglund, J.; Markström, K., Generation and properties of snarks, J. Combin. Theory, Ser. B, 103, 468-488 (2013) · Zbl 1301.05119
[6] Brinkmann, G.; McKay, B. D., Construction of planar triangulations with minimum degree 5, Discrete Math., 301, 147-163 (2005) · Zbl 1078.05022
[7] Brinkmann, G.; McKay, B. D., Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58, 323-357 (2007) · Zbl 1164.68025
[8] Chartrand, G.; Gould, R. J.; Kapoor, S. F., On homogeneously traceable nonhamiltonian graphs, Ann. N. Y. Acad. Sci., 319, 130-135 (1979) · Zbl 0481.05039
[9] Clark, L.; Entringer, R., Smallest maximally nonhamiltonian graphs, Period. Math. Hung., 14, 57-68 (1983) · Zbl 0489.05038
[10] Coxeter, H. S.M., Self-dual configurations and regular graphs, Bull. Amer. Math. Soc., 56, 413-455 (1950) · Zbl 0040.22803
[11] Ferrero, D.; Hanusch, S., Component connectivity of generalized Petersen graphs, Internat. J. Comput. Math., 91, 1940-1963 (2014) · Zbl 1408.05077
[12] J. Goedgebeur, C.T. Zamfirescu, Homepage of a generator for hypohamiltonian graphs, 2016, http://caagt.ugent.be/hypoham/.
[13] Goedgebeur, J.; Zamfirescu, C. T., Improved bounds for hypohamiltonian graphs, Ars Math. Contemp., 13, 235-257 (2017) · Zbl 1380.05034
[14] Goedgebeur, J.; Zamfirescu, C. T., On hypohamiltonian snarks and a theorem of Fiorini, Ars Math. Contemp., 14, 227-249 (2018) · Zbl 1395.05044
[15] Holton, D. A.; McKay, B. D., The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combin. Theory, Ser. B, 45, 305-319 (1988) · Zbl 0607.05051
[16] Holton, D. A.; Sheehan, J., The Petersen graph, chapter 7: Hypohamiltonian graphs (1993), Cambridge University Press: Cambridge University Press New York · Zbl 0781.05001
[17] Máčajová, E.; Škoviera, M., Infinitely many hypohamiltonian cubic graphs of girth 7, Graphs Combin., 27, 231-241 (2011) · Zbl 1235.05085
[18] B.D. McKay, nauty user’s guide (version 2.6), 2017. Technical Report TR-CS-90-02, Department of Computer Science, Australian National University. The latest version of the software is available at http://cs.anu.edu.au/ bdm/nauty.
[19] McKay, B. D.; Piperno, A., Practical graph isomorphism, II, J. Symbolic Comput., 60, 94-112 (2014) · Zbl 1394.05079
[20] M.Sc. thesis (in Dutch).
[21] Watkins, M. E., A theorem on Tait colorings with an application to generalized Petersen graphs, J. Combin. Theory, 6, 152-164 (1969) · Zbl 0175.50303
[22] Zamfirescu, C. T., On non-hamiltonian graphs for which every vertex-deleted subgraph is traceable, J. Graph Theory, 86, 223-243 (2017) · Zbl 1370.05115
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