Discussiones Mathematicae Graph Theory 33(1) (2013)
71-89
DOI: https://doi.org/10.7151/dmgt.1653
Dedicated to Mietek Borowiecki on the occasion of his 70th birthday.
On Vertices Enforcing a Hamiltonian Cycle
Igor Fabrici
Institute of Mathematics | Erhard Hexel
Institut für Mathematik | Stanislav Jendrol'
Institut of Mathematics |
Abstract
A nonempty vertex set X ⊆ V(G) of a hamiltonian graph G is called an of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.
Keywords: cycle, hamiltonian, 1-hamiltonian
2010 Mathematics Subject Classification: 05C45.
References
[1] | C.A. Barefoot, Hamiltonian connectivity of the Halin graphs, Congr. Numer. 58 (1987) 93--102. |
[2] | J.A. Bondy, Pancyclic graphs: recent results, in: Infinite and finite sets, Vol. 1, Colloq. Math. Soc. János Bolyai 10, ed(s), A. Hajnal, R. Rado and V.T. Sós North Holland, 1975) 181--187. |
[3] | J.A. Bondy and L. Lovász, Cycles through specified vertices of a graph, Combinatorica 1 (1981) 117--140, doi: 10.1007/BF02579268. |
[4] | H.J. Broersma and H.J. Veldman, 3-connected line graphs of triangular graphs are panconnected and 1-hamiltonian, J. Graph Theory 11 (1987) 399--407, doi: 10.1002/jgt.3190110314. |
[5] | G. Chartrand, A.M. Hobbs, H.A. Jung, S.F. Kapoor, D.R. Link and C.St.J.A. Nash-Williams, The square of a block is hamiltonian connected, J. Combin. Theory. (B) 16 (1974) 290--292, doi: 10.1016/0095-8956(74)90075-6. |
[6] | G. Chartrand, S.F. Kapoor and D.R. Link, n-hamiltonian graphs, J. Combin. Theory 9 (1970) 308--312, doi: 10.1016/S0021-9800(70)80069-2. |
[7] | G. Chartrand and L. Lesniak, Graphs & Digraphs (Chapman & Hall, 2005). |
[8] | N. Chiba and T. Nishizeki, A theorem on paths in planar graphs, J. Graph Theory 10 (1986) 449--450, doi: 10.1002/jgt.3190100404 . |
[9] | V. Chvátal and P. Erdös, A note on hamiltonian circuits, Discrete Math. 2 (1972) 111--113, doi: 10.1016/0012-365X(72)90079-9. |
[10] | G.A. Dirac, In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen, Math. Nachr. 22 (1960) 61--85, doi: 10.1002/mana.19600220107. |
[11] | P. Erdös and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959) 337--356, doi: 10.1007/BF02024498. |
[12] | H. Fleischner, The square of every two-connected graph is hamiltonian, J. Combin. Theory (B) 16 (1974) 29--34, doi: 10.1016/0095-8956(74)90091-4. |
[13] | R. Gould, A look at cycles containing specified elements of a graph, Discrete Math. 309 (2009) 6299--6311, doi: 10.1016/j.disc.2008.04.017. |
[14] | S.V. Kanetkar and P.R. Rao, Connected, locally 2-connected, K1,3-free graphs are panconnected, J. Graph Theory 8 (1984) 347--353, doi: 10.1002/jgt.3190080302. |
[15] | K. Kawarabayashi, Cycles through a prescribed vertex set in n-connected graph, J. Combin. Theory B 90 (2004) 315--323, doi: 10.1016/j.jctb.2003.08.002. |
[16] | L. Lovász and M.D. Plummer, On a family of planar bicritical graphs, Proc. London Math. Soc. 30 (1975) 160--176, doi: 10.1112/plms/s3-30.2.160. |
[17] | D.A. Nelson, Hamiltonian graphs, Master Thesis (Vanderbilt University, 1973). |
[18] | D.J. Oberly and D.P. Sumner, Every connected, locally connected nontrivial graph with no induced claw is hamiltonian, J. Graph Theory 3 (1979) 351--356, doi: 10.1002/jgt.3190030405. |
[19] | O. Ore, Hamilton connected graphs, J. Math. Pures Appl. 42 (1963) 21--27. |
[20] | C. Thomassen, A theorem on paths in planar graphs, J. Graph Theory 7 (1983) 169--176, doi: 10.1002/jgt.3190070205. |
[21] | W.T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956) 99--116, doi: 10.1090/S0002-9947-1956-0081471-8. |
[22] | T. Zamfirescu, Three small cubic graphs with interesting hamiltonian properties, J. Graph Theory 4 (1980) 287--292, doi: 10.1002/jgt.3190040306. |
Received 19 April 2012
Accepted 25 October 2012
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