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Hamiltonian properties of triangular grid graphs. (English) Zbl 1158.05040
Summary: A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, J.R. Reay and T. Zamfirescu [Discrete Comput. Geom. 24, No. 2-3, 497–502 (2000; Zbl 0953.05040)] showed that all 2-connected, linearly-convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph $$D$$ which is the linearly-convex hull of the Star of David. We extend this result to a wider class of locally connected triangular grid graphs. Namely, we prove that all connected, locally connected triangular grid graphs (with the same exception of graph $$D$$) are hamiltonian. Moreover, we present a sufficient condition for a connected graph to be fully cycle extendable. We also show that the problem Hamiltonian Cycle is NP-complete for triangular grid graphs.

##### MSC:
 05C45 Eulerian and Hamiltonian graphs 92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.) 92C40 Biochemistry, molecular biology
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##### References:
 [1] Agarwala, R.; Batzoglou, S.; Dančík, V.; Decatur, S.E.; Farach, M.; Hannenhalli, S.; Skiena, S., Local rules for protein folding on a triangular lattice and generalized hydrophobicity in the HP model, J. comput. biology, 4, 275-296, (1997) · Zbl 1321.92051 [2] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), Macmillan London, Elsevier, New York · Zbl 1134.05001 [3] Chartrand, G.; Pippert, R., Locally connected graphs, Casopis pest. mat., 99, 158-163, (1974) · Zbl 0278.05113 [4] Clark, L., Hamiltonian properties of connected locally connected graphs, Congr. numer., 32, 199-204, (1981) [5] des Cloizeaux, J.; Jannik, G., Polymers in solution: their modelling and structure, (1987), Clarendon Press Oxford [6] Faudree, R.J.; Flandrin, E.; Ryjáček, Z., Claw-free graphs — A survey, Discrete math., 164, 87-147, (1997) · Zbl 0879.05043 [7] Garey, M.R.; Johnson, D.S., Computers and intractability. A guide to the theory of NP-completeness, (1979), W.H. Freeman and Co. San Francisco, CA · Zbl 0411.68039 [8] Grünbaum, B.; Shephard, G.C., Tilings and patterns: an introduction, (1989), W.H. Freeman New York · Zbl 0746.52001 [9] Havet, F., Channel assignment and multicoloring of the induced subgraphs of the triangular lattice, Discrete math., 233, 219-231, (2001) · Zbl 0983.05031 [10] Hendry, G.R.T., A strengthening of kikust’s theorem, J. graph theory, 13, 257-260, (1989) · Zbl 0668.05042 [11] Hendry, G.R.T., Extending cycles in graphs, Discrete math., 85, 59-72, (1990) · Zbl 0714.05038 [12] Itai, A.; Papadimitriou, C.H.; Szwarcfiter, J.L., Hamiltonian paths in grid graphs, SIAM J. comput., 11, 676-686, (1982) · Zbl 0506.05043 [13] Lenhart, W.; Umans, C., Hamiltonian cycles in solid grid graphs, (), 496-505 [14] Lua, R.; Borovinskiy, A.L.; Grosberg, A.Yu., Fractal and statistical properties of large compact polymers: A computational study, Polymer, 45, 717-731, (2004) [15] Oberly, D.J.; Sumner, D.P., Every connected, locally connected nontrivial graph with no induced claw is Hamiltonian, J. graph theory, 3, 351-356, (1979) · Zbl 0424.05036 [16] Orlovich, Yu.L., Kikust – hendry type conditions for Hamiltonian graphs, Tr. inst. mat., 3, 149-156, (1999), (in Russian) · Zbl 0949.05050 [17] Orlovich, Yu.L., On locally connected graphs whose maximal vertex degree is at most four, Vestsi nats. akad. navuk belarusi. ser. fiz.-mat. navuk, 3, 97-100, (1999), (in Russian) [18] Orlovich, Yu.L.; Gordon, V.S.; Werner, F., Hamiltonian cycles in graphs of triangular grid, Doklady NASB, 49, 5, 21-25, (2005), (in Russian) · Zbl 1178.05057 [19] Orlovich, Yu.; Gordon, V.; Werner, F., Cyclic properties of triangular grid graphs, (), 149-153 [20] Plesnik, J., The NP-completeness of the Hamiltonian cycle problem in bipartite cubic planar graphs, Acta math. univ. Comenian., 42-43, 271-273, (1983) · Zbl 0539.05043 [21] Reay, J.R.; Zamfirescu, T., Hamiltonian cycles in $$T$$-graphs, Discrete comput. geom., 24, 497-502, (2000) · Zbl 0953.05040
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