Year 2015,
, 111 - 116, 30.04.2015
Abstract
A graph $G$ is said to be \textit{$4$-ordered} if for any ordered set of four distinct vertices of $G$, there exists a cycle in $G$ that contains all of the four vertices in the designated order. Furthermore, if we can find such a cycle as a Hamiltonian cycle, $G$ is said to be \textit{$4$-ordered Hamiltonian}. It was shown that every $4$-connected planar triangulation is (i) Hamiltonian (by Whitney) and (ii) $4$-ordered (by Goddard). Therefore, it is natural to ask whether every $4$-connected planar triangulation is $4$-ordered Hamiltonian. In this paper, we give a partial solution to the problem, by showing that every $5$-connected planar triangulation is $4$-ordered Hamiltonian.
References
- D. Archdeacon, N. Hartsfield, C. H. C. Little,Nonhamiltonian triangulations with large connectivity and representativity, J. Combin. Theory Ser. B, 68, 45-55, 1996.
- R. J. Faudree,Survey of results on k-ordered graphs, Discrete Math., 229, 73-87, 2001.
- W. Goddard, 4-connected maximal planar graphs are 4-ordered, Discrete Math., 257, 405-410, 2002.
- J. W. Moon and L. Moser,Simple paths on polyhedra, Pacific J. Math., 13, 629-631, 1963.
- R. Mukae and K. Ozeki, 4-connected triangulations and 4-orderedness, Discrete Math.,310, 2271- 2272, 2010.
- K. Kawarabayashi and K. Ozeki,4-connected projective planar graphs are hamiltonian-connected, (to appear in) J. Combin. Theory Ser. B.
- D. P. Sanders,On paths in planar graphs, J. Graph Theory, 24, 341-345, 1997.
- R. Thomas and X. Yu, 4-connected projective-planar graphs are Hamiltonian, J. Combin. Theory Ser. B, 62, 114-132, 1994.
- C. Thomassen,A theorem on paths in planar graphs, J. Graph Theory, 7, 169-176, 1983.
- C. Thomassen,Trees in triangulations, J. Combin. Theory Ser. B, 60, 58-62, 1994.
- W. T. Tutte,A theorem on planar graphs, Trans. Amer. Math. Soc., 82, 99-116, 1956.
- H. Whitney,A theorem on graphs, Ann. of Math., 32, 378-390, 1931.
- X. Yu,Disjoint paths, planarizing cycles, and spanning walks, Trans. Amer. Math. Soc., 349, 1333- 1358, 1997.
Year 2015,
, 111 - 116, 30.04.2015
Abstract
References
- D. Archdeacon, N. Hartsfield, C. H. C. Little,Nonhamiltonian triangulations with large connectivity and representativity, J. Combin. Theory Ser. B, 68, 45-55, 1996.
- R. J. Faudree,Survey of results on k-ordered graphs, Discrete Math., 229, 73-87, 2001.
- W. Goddard, 4-connected maximal planar graphs are 4-ordered, Discrete Math., 257, 405-410, 2002.
- J. W. Moon and L. Moser,Simple paths on polyhedra, Pacific J. Math., 13, 629-631, 1963.
- R. Mukae and K. Ozeki, 4-connected triangulations and 4-orderedness, Discrete Math.,310, 2271- 2272, 2010.
- K. Kawarabayashi and K. Ozeki,4-connected projective planar graphs are hamiltonian-connected, (to appear in) J. Combin. Theory Ser. B.
- D. P. Sanders,On paths in planar graphs, J. Graph Theory, 24, 341-345, 1997.
- R. Thomas and X. Yu, 4-connected projective-planar graphs are Hamiltonian, J. Combin. Theory Ser. B, 62, 114-132, 1994.
- C. Thomassen,A theorem on paths in planar graphs, J. Graph Theory, 7, 169-176, 1983.
- C. Thomassen,Trees in triangulations, J. Combin. Theory Ser. B, 60, 58-62, 1994.
- W. T. Tutte,A theorem on planar graphs, Trans. Amer. Math. Soc., 82, 99-116, 1956.
- H. Whitney,A theorem on graphs, Ann. of Math., 32, 378-390, 1931.
- X. Yu,Disjoint paths, planarizing cycles, and spanning walks, Trans. Amer. Math. Soc., 349, 1333- 1358, 1997.
There are 13 citations in total.
Details
| Primary Language | English |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | April 30, 2015 |
| Published in Issue | Year 2015 |