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2013 | 33 | 1 | 133-137
Tytuł artykułu

A Note on Barnette’s Conjecture

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices.
Wydawca
Rocznik
Tom
33
Numer
1
Strony
133-137
Opis fizyczny
Daty
wydano
2013-03-01
online
2013-04-13
Twórcy
Bibliografia
  • [1] D. Barnette, Conjecture 5, Recent Problems in Combinatorics, W.T. Tutte, (Ed.), Academic Press, New York, 1969, p. 343.
  • [2] J.A. Bondy and S.C. Locke, Relative lengths of paths and cycles in 3-connected graphs, Discrete Math. 33 (1981) 111-122. doi:10.1016/0012-365X(81)90159-X[WoS][Crossref]
  • [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan Co., New York, 1976).
  • [4] R. Diestel, Graph Theory, Springer, Graduate Texts in Mathematics 173(2000).
  • [5] M.N. Ellingham and J.D. Horton, Non-hamiltonian 3-connected cubic bipartite graphs, J. Combin. Theory (B) 34 (1983) 330-333. doi:10.1016/0095-8956(83)90046-1[Crossref]
  • [6] B. Grünbaum, Polytopes, graphs and complexes, Bull. Amer. Math. Soc. 76 (1970) 1131-1201. doi:10.1090/S0002-9904-1970-12601-5[Crossref]
  • [7] A. Hertel, Hamiltonian cycles in sparse graphs, Masters Thesis, University of Toronto, 2004.
  • [8] A. Hertel, A survey & strengthening of Barnette’s Conjecture, University of Toronto, 2005.
  • [9] J.D. Horton, A counterexample to Tutte’s conjecture, in [3], p. 240.
  • [10] T.R. Jensen and B. Toft, Graph Coloring Problems, J. Wiley & Sons ( New York, 1995) page 45. doi:10.1002/9781118032497[Crossref]
  • [11] A.K. Kelmans, Constructions of cubic bipartite and 3-connected graphs without Hamiltonian cycles, Analiz Zadach Formirovaniya i Vybora Alternativ, VNIISI, Moscow 10 (1986) 64-72, in Russian. (see also AMS Translations, Series 2 158 (1994) 127-140, A.K. Kelmans, (Ed.))
  • [12] A.K. Kelmans, Graph planarity and related topics, Contemp. Math. 147 (1993) 635-667. doi:10.1090/conm/147/01205[Crossref]
  • [13] A.K. Kelmans, Konstruktsii kubicheskih dvudolnyh 3-Svyaznyh bez Gamiltonovyh tsiklov, Sb. Tr. VNII Sistem. Issled. 10 (1986) 64-72.
  • [14] A.K. Kelmans, Kubicheskie dvudolnye tsiklicheski 4-Svyaznye grafy bez Gamiltonovyh tsiklov, Usp. Mat. Nauk 43(3) (1988) 181-182.
  • [15] M. Król, On a sufficient and necessary condition of 3-colorability of a planar graph, I, Prace Nauk. Inst. Mat. Fiz. Teoret. 6 (1972) 37-40.
  • [16] M. Król, On a sufficient and necessary condition of 3-colorability of a planar graph, II, Prace Nauk. Inst. Mat. Fiz. Teoret. 9 (1973) 49-54.
  • [17] S.K. Stein, B-sets and coloring problems, Bull. Amer. Math. Soc. 76 (1970) 805-806. doi:10.1090/S0002-9904-1970-12559-9[Crossref]
  • [18] S.K. Stein, B-sets and planar maps, Pacific. J. Math. 37 (1971) 217-224.
  • [19] P.G. Tait, Listings topologie, Phil. Mag. 17 (1884) 30-46.
  • [20] W.T. Tutte, On Hamiltonian circuits, J. London Math. Soc. 21 (1946) 98-101. doi:10.1112/jlms/s1-21.2.98[Crossref]
  • [21] W.T. Tutte, On the 2-factors of bicubic graphs, Discrete Math. 1 (1971) 203-208. doi:10.1016/0012-365X(71)90027-6[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1643
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