Abstract
This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains some material on closely related topics such as traceable, pancyclic and Hamiltonian connected graphs.
Similar content being viewed by others
References
Abderrezzak M.E.K., Flandrin E., Amar D.: Cyclability and pancyclability in bipartite graphs. Discrete Math. 236, 3–11 (2001)
Abueida A., Sritharan R.: Cycle extendability and Hamiltonian cycles in chordal graph classes. SIAM J. Discrete Math. 20, 669–681 (2006)
Abueida, A., Busch, A., Sritharan, R.: Hamiltonian spider intersection graphs are cycle extendable (preprint)
Ainouche A.: Dirac’s type sufficient conditions for Hamiltonicity and pancyclicity. Graphs Combin. 25, 129–137 (2009)
Ainouche A.: Extensions of Bondy’s theorem on cycles in 2-connected graphs. Ars Combin. 85, 385–393 (2007)
Ainouche A., Broersma H.J., Veldman H.J.: Remarks on Hamiltonian properties of claw-free graphs. Ars Combin. 29, 110–121 (1990)
Ainouche A., Kouider M.: Hamiltonism and partially square graphs. Graphs Combin. 15, 257–265 (1999)
Ainouche A., Lapiquonne S.: Hamiltonian connectedness and partially square graphs. Discrete Math. 306, 1097–1104 (2006)
Ainouche A., Schiermeyer I.: 0-Dual closures for several classes of graphs. Graphs Combin. 19, 297–307 (2003)
Alspach B.: Research problem 59. Discrete Math. 50, 115 (1984)
Alspach B.: The wonderful Walecki construction. Bull. Inst. Combin. Appl. 52, 7–20 (2008)
Alspach B., Bryant D., Dyer D.: Paley graphs have Hamilton decompositions. Discrete Math. 312, 113–118 (2012)
Bailey R.F., Stevens B.: Hamiltonian decompositions of complete k-uniform hypergraphs. Discrete Math. 310, 3088–3095 (2010)
Babu Ch.S., Diwan A.A.: Subdivisions of graphs: a generalization of paths and cycles. Discrete Math. 308, 4479–4486 (2008)
Bal D., Frieze A.: Packing tight Hamilton cycles in uniform hypergraphs. SIAM J. Discrete Math. 26, 435–451 (2012)
Balogh J., Bollobás B., Krivelevich M., Müller T., Walters M.: Hamilton cycles in random geometric graphs. Ann. Appl. Probab. 21, 1053–1072 (2011)
Balakrishnan R., Bermond J.-C., Paulraja P., Yu M.-L.: On Hamilton cycle decompositions of the tensor product of complete graphs. Discrete Math. 268, 49–58 (2003)
Barnette, D.: Conjecture 5. In: Tutte, W. (ed.) Recent Progress in Combinatorics, p. 343. Academic Press, New York (1969)
Bauer D., Schmeichel E.: Binding number, minimum degree, and cycle structure in graphs. J. Graph Theory 71, 219–228 (2012)
Bedrosian, P.: Forbidden subgraphs and minimum degree conditions for Hamiltonicity. Ph.D. Thesis, Memphis State University (1991)
Ben-Shimon S., Krivelevich M., Sudakov B.: On the resilience of hamiltonicity and optimal packing of Hamiltonian cycles in random graphs. SIAM J. Discrete Math. 25, 1176–1193 (2011)
Bermond J.-C.: Hamiltonian decompositions of graphs, directed graphs and hypergraphs. Advances in Graph Theory (Cambridge Combinatorial Conference, Trinity College, Cambridge, 1977). Ann. Discrete Math. 3, 21–28 (1978)
Bermond J.-C.: Problem 97. Discrete Math. 71, 275 (1988)
Bermond J.-C., Germa A., Heydemann M.C., Sotteau D.: Hypergraphes hamiltoniens. Prob. Comb. Théorie Graph Orsay 260, 39–43 (1976)
Bian Q., Horn P., Janiszewski S., La Fleur S., Gould R.J., Wrayno P.: 3-connected {K 1,3, P 9 }-free graphs are Hamiltonian connected. Graphs Combin. 313, 2772–2777 (2013)
Biebighauser D.P., Ellingham M.N.: Prism-Hamiltonicity of triangulations. J. Graph Theory 57, 181–197 (2008)
Bloznelis M., Radavičius I.: A note on Hamiltonicity of uniform random intersection graphs. Lith. Math. J. 51, 155–161 (2011)
Bohman T., Frieze A.: Hamilton cycles in 3-out. Random Struct. Algorithms 35, 393–417 (2009)
Böhme T., Harant J., Tkáč M.: More than one tough chordal planar graphs are Hamiltonian. J. Graph Theory 32, 405–410 (1999)
Bondy J.A.: Pancyclic graphs I. J. Combin. Theory Ser. B 11, 80–84 (1971)
Böttcher J., Kohayakawa Y., Procacci A.: Properly colored copies and rainbow copies of large graphs with small maximum degree. Random Struct. Algorithms 40, 425–436 (2012)
Broersma H., Kriesell M., Ryjáček Z.: On factors of 4-connected claw-free graphs. J. Graph Theory 37, 125–136 (2001)
Broersma H., Faudree R.J., Huck A., Trommel H., Veldman H.J.: Forbidden subgraphs that imply Hamiltonian connectedness. J. Graph Theory 40, 104–119 (2002)
Broersma H., Ryjacek Z., Vrána P.: How many conjectures can you stand? A survey. Graphs Combin. 28, 57–75 (2012)
Broersma H., Xiong L., Yoshimoto K.: Toughness and Hamiltonicity in k-trees. Discrete Math. 307, 832–838 (2007)
Brousek J.: Forbidden triples for Hamiltonicity. Discrete Math. 251, 71–76 (2002)
Brunet, R., Nakamoto, A., Negami, S.: Every 5-connected triangulation of the Klein bottle is Hamiltonian. In: Proceedings of the 10th Workshop on Topological Graph Theory (Yokohama, 1998) Yokohama Mathematical Journal, vol. 47, pp. 239–244 (1999)
Bryant D., Leach C.D., Rodger C.: Hamilton decompositions of complete bipartite graphs with 3-factor leaves. Australas. J. Combin. 31, 331–336 (2005)
Butler S., Chung F.: Small spectral gap in the combinatorial Laplacian implies Hamiltonian. Ann. Comb. 13, 403–412 (2010)
Čada R., Flandrin E., Li H., Ryjáček Z.: Cycles through given vertices and closures. Discrete Math. 276, 65–80 (2004)
Chen C.: Any maximal planar graph with only one separating triangle is Hamiltonian. J. Comb. Optim. 7, 79–86 (2003)
Chen B., Zhang S., Qiao Q.: Hamilton cycles in claw-heavy graphs. Discrete Math. 309, 2015–2019 (2009)
Chen G., Fan G., Yu X.: Cycles in 4-connected planar graphs. Eur. J. Combin. 25, 763–780 (2004)
Chen G., Faudree R.J., Gould R.J., Jacobson M.S.: Cycle extendability of Hamiltonian interval graphs. SIAM J. Discrete Math. 20, 682–689 (2006)
Chen G., Faudree R.J., Gould R.J., Jacobson M.S., Lesniak L., Pfender F.: Linear forests and ordered cycles. Discussiones Mathematicae Graph Theory 24, 359–372 (2004)
Chen G., Gould R.J.: Hamiltonian connected graphs involving forbidden subgraphs. Bull. Inst. Combin. Appl. 29, 25–32 (2000)
Chen G., Gould R.J., Pfender F.: New conditions for k-ordered Hamiltonian graphs. Ars Combin. 70, 245–255 (2004)
Christofides D., Kühn D., Osthus D.: Edge-disjoint Hamiltonian cycles in graphs. J. Combin. Theory Ser. B 102, 1035–1060 (2012)
Chvátal V.: Tough graphs and Hamiltonian circuits. Discrete Math. 5, 215–223 (1973)
Chvátal V., Erdös P.: A note on Hamiltonian circuits. Discrete Math. 2, 111–113 (1972)
Cooper C., Frieze A., Krivelevich M.: Hamilton graphs in random graphs with a fixed degree sequence. SIAM J. Discrete Math. 24, 558–569 (2010)
Cuckler B., Kahn J.: Hamiltonian cycles in Dirac graphs. Combinatorica 29, 299–326 (2009)
Cui Q., Hu Y., Wang J.: Long cycles in 4-connected planar graphs. Discrete Math. 309, 1051–1059 (2009)
Diaz J., Mitsche D., Pérez X.: Sharp threshold for Hamiltonicity of random geometric graphs. SIAM J. Discrete Math. 21, 57–65 (2007)
Dirac G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)
Dudek, A., Ferrara, M.: Extensions of results on rainbow Hamilton cycles in uniform hypergraphs (preprint)
Dudek, A., Frieze, A.: Loose Hamiltonian cycles in random uniform hypergraphs. Electron. J. Combin. 18 (2011), Paper 48
Dudek A., Frieze A.: Tight Hamilton cycles in random uniform hypergraphs. Random Struct. Algorithms 42, 374–385 (2012)
Dudek, A., Frieze, A., Loh, P.-S., Speiss, S.: Optimal divisibility conditions for loose Hamilton cycles in random hypergraphs. Electron. J. Combin. 19 (2012), Paper 44
Dudek, A., Frieze, A., Rucinski, A.: Rainbow Hamilton cycles in uniform hypergraphs. Electron. J. Combin. 19 (2012), Paper 46
Duffus, D., Gould, R.J., Jacobson, M.S.: Forbidden subgraphs and the Hamiltonian theme. In: Chartrand, Alavi, Goldsmith, Lesniak, Lick (eds.) The Theory and Applications of Graphs, (Kalmazoo, Mich., 1980), pp. 297–316. Wiley, New York (1981)
Egawa Y., Glas R., Locke S.: Cycles and paths through specified vertices in k-connected graphs. J. Combin. Theory Ser. B 52, 20–29 (1991)
Efthymiou, C., Spitakas, P.G.: On the existence of Hamiltonian cycles in random intersection graphs. Lecture Notes in Computer Science, pp. 690–701. Springer, Berlin (2005)
Efthymiou C., Spirakas P.G.: Sharp thresholds for hamiltonicity in random intersection graphs. Theor. Comput. Sci. 411, 3714–3730 (2010)
Enomoto, H.: Personal communication
Fan G.H.: New sufficient condition for cycles in graphs. J. Combin. Theory B 37, 221–227 (1984)
Faudree J.R., Faudree R.J.: Hamiltonian cycles containing ordered linear forests. Bull. Inst. Combin. Appl. 5, 78–104 (2009)
Faudree J.R., Faudree R.J., Gould R.J., Jacobson M.S., Lesniak L.: On k-ordered graphs. J. Graph Theory 35, 69–82 (2000)
Faudree J.R., Faudree R.J., Ryjáčk Z., Vrána P.: On forbidden pairs implying Hamilton-connectedness. J. Graph Theory 72, 327–345 (2013)
Faudree R.J., Gould R.J.: Characterizing forbidden pairs for Hamiltonian properties. Discrete Math. 173, 45–60 (1997)
Faudree R.J., Gould R.J., Jacobson M.J., Lesniak L.: Characterizing forbidden clawless triples implying Hamiltonian graphs. Discrete Math. 249, 71–81 (2002)
Faudree R.J., Gould R.J., Jacobson M.S.: Forbidden triples implying hamiltonicity: for all graphs. Discuss. Math. Graph Theory 24, 47–54 (2004)
Faudree R.J., Gould R.J., Jacobson M.S.: Potential forbidden triples implying Hamiltonicity: for sufficiently large graphs. Discuss. Math. Graph Theory 25, 273–289 (2005)
Faudree R.J., Gould R.J., Jacobson M.S., Lesniak L.: Minimum degree and (k,m)-pancyclic ordered graphs. Graphs Combin. 21, 197–211 (2005)
Faudree R.J., Gould R.J., Kostochka A., Lesniak L., Schiermeyer I., Saito A.: Degree conditions for k-ordered Hamiltonian graphs. J. Graph Theory 42, 199–210 (2003)
Faudree, R.J., Gould, R.J., Jacobson, M.S.: Pancyclic graphs and linear forests. Discrete Math. (2013, to appear)
Faudree R.J., Gould R.J., Jacobson M.S., Magnant C.: Distributing vertices on Hamiltonian cycles. J. Graph Theory 69, 28–45 (2012)
Faudree, R.J., Gould, R.J.: Precise location of vertices on Hamiltonian cycles. Discrete Math. 313(23), 2772–2777 (2013)
Faudree R.J., Gould R.J., Jacobson M.S., Lesniak L.: Generalizing pancyclic and k-ordered graphs. Graphs Combin. 20, 291–309 (2004)
Faudree, R.J., Lehel, J., Yoshimoto, K.: A note on locating pairs of vertices on a Hamiltonian cycle (preprint)
Favaron O., Flandrin E., Li H., Tian F.: An Ore-type condition for pancyclability. Discrete Math. 206, 139–144 (1999)
Ferrara M., Gould R.J., Jacobson M.S., Pfender F., Powell J., Whalen T.: New Ore-type conditions for H-linked graphs. J. Graph Theory 71, 69–77 (2012)
Ferrara M., Gould R.J., Tansey G., Whalen T.: Disjoint Hamiltonian cycles in bipartite graphs. Discrete Math. 309, 3811–3820 (2009)
Ferrara M., Gould R.J., Tansey G., Whalen T.: On H-linked graphs. Graphs Combin. 22, 217–224 (2006)
Ferrara M., Jacobson M.S., Harlan A.: Hamiltonian cycles avoiding sets of edges in a graph. Australas. J. Combin. 48, 191–203 (2010)
Ferrara M., Magnant C., Powell J.: Pan-H-linked graphs. Graphs Combin. 26, 225–242 (2010)
Fiedler M., Nikiforov V.: Spectral radius and hamiltonicity of graphs. Linear Algebra Appl. 432, 2170–2173 (2010)
Flandrin E., Li H., Marczyk A., Wožniak M.: A note on pancyclism of highly connected graphs. Discrete Math. 286, 57–60 (2004)
Florek J.: On Barnette’s conjecture. Discrete Math. 310, 1531–1535 (2010)
Frankl P., Katona Gy.Y.: Extremal k-edge Hamiltonian hypergraphs. Discrete Math. 308, 1415–1424 (2008)
Frieze, A.: Loose Hamilton cycles in random 3-uniform hypergraphs. Electron. J. Combin. 17, #N283 (2010)
Frieze A., Krivelevich M.: On packing Hamilton cycles in ε-regular graphs. J. Combin. Theory B 94, 159–172 (2005)
Frieze A., Krivelevich M., Loh P.-S.: Packing tight Hamilton cycles in 3-uniform hypergraphs. Random Struct. Algorithms 40, 269–300 (2012)
Fujisawa J., Ota K., Sugiyama T., Tsugaki M.: Forbidden subgraphs and the existence of paths and cycles through specified vertices. Discrete Math. 308, 6111–6114 (2008)
Fujisawa J., Nakamoto A., Ozeki K.: Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices. J. Combin. Theory B 103, 46–60 (2013)
Fujisawa J., Yamashita T.: Degree conditions on claws and modified claws for hamiltonicity of graphs. Discrete Math. 308, 1612–1619 (2008)
Gerlach T.: Toughness and hamiltonicity of a class of planar graphs. Discrete Math. 286, 61–65 (2004)
Glebov R., Krivelevich M.: On the number of Hamilton cycles in sparse random graphs. SIAM J. Discrete Math. 27, 27–42 (2013)
Glebov R., Person Y., Weps W.: On extremal hypergraphs for Hamiltonian cycles. Eur. J. Combin. 33, 544–555 (2012)
Goddard W.: Minimum degree conditions for cycles including specified sets of vertices. Graphs Combin. 20, 467–483 (2004)
Gould R.J.: Updating the Hamiltonian problem—a survey. J. Graph Theory 15, 121–157 (1991)
Gould R.J.: Advances on the Hamiltonian problem—a survey. Graphs Combin. 19, 7–52 (2003)
Gould R.J.: A look at cycles containing specified elements of a graph. Discrete Math. 309, 6299–6311 (2009)
Gould, R.J.: Graph Theory, Dover Publications, Inc., Mineola (2012)
Gould R.J., Luczak T., Pfender F.: Pancyclicity of 3-connected graphs: pairs of forbidden subgraphs. J. Graph Theory 47, 183–202 (2004)
Gould R.J., Kostochka A., Yu G.: On minimum degree implying that a graph is H-linked. SIAM J. Discrete Math. 20, 829–840 (2006)
Gould R.J., Whalen T.: Subdivision extendibility. Graphs Combin. 23, 165–182 (2007)
Greenhill C., Kim J.H., Wormald N.: Hamiltonian decompositions of random bipartite regular graphs. J. Combin. Theory Ser. B 90, 195–222 (2004)
Grünbaum B.: Polytopes graphs and complexes. Bull. Am. Math. Soc. 76, 1131–1201 (1970)
Häggkvist, R.: On F-Hamiltonian graphs. In: Bondy, J.A., Murty, U.S.R. (eds) Graphs and Related Topics, pp. 219–231. Academic Press, New York (1979)
Hán H., Schacht M.: Dirac-type results for loose Hamilton cycles in uniform hypergraphs. J. Combin. Theory B 100, 332–346 (2010)
Harkat-Benhamdine A., Li H., Tian F.: Cyclability of 3-connected graphs. J. Graph Theory 34, 191–203 (2000)
Hartke, S., Seacrest, T.: Random partitions and edge disjoint Hamiltonian cycles (preprint)
Helden G.: Each maximal planar graph with exactly two separating triangles is Hamiltonian. Discrete Appl. Math. 155, 1833–1836 (2007)
Helden, G., Vieten, O.: Hamiltonian cycles in maximal planar graphs. In: Cologne-Twente Workshop on Graphs and Combinatorial Optimization 71, Electronic Notes in Discrete Mathematics, vol. 25, Elsevier, Amsterdam (2006)
Holton D.L., Manvel B., McKay B.D.: Hamiltonian cycles in cubic 3-connected bipartite graphs. J. Combin. Theory B 38, 279–297 (1985)
Hu Z., Tian F., Bing W.: Hamiltonian connectivity of line graphs and claw-free graphs. J. Graph Theory 50, 130–141 (2005)
Jackson B.: Edge-disjoint Hamiltonian cycles in regular graphs of large degree. J. Lond. Math. Soc. 19, 13–16 (1979)
Jiang T.: Planar Hamiltonian chordal graphs are cycle extendable. Discrete Math. 257, 441–444 (2002)
Kaiser T., Kriesell M.: On the pancyclicity of lexicographic products. Graphs Combin. 22, 51–58 (2006)
Kaiser T., Vrána P.: Hamilton cycles in 5-connected line graphs. Eur. J. Combin. 33, 924–947 (2012)
Kaneko A., Yoshimoto K.: On a Hamiltonian cycle in which specified vertices are uniformly distributed. J. Combin. Theory Ser. B 81, 100–109 (2001)
Karoński M., Scheinerman E., Singer-Cohen K.: On random intersection graphs: the subgraph problem. Combin. Prob. Comput. 8, 131–159 (1999)
Katona G.Y., Kierstead H.: Hamiltonian chains in hypergraphs. J. Graph Theory 30, 205–212 (1999)
Kawarabayashi K.: A survey on Hamiltonian cycles. Interdiscip. Inf. Sci. 7, 25–39 (2001)
Keevash P.: A hypergraph blow-up lemma. Random Struct. Algorithms 39, 275–376 (2013)
Keevash P., Kühn D., Mycroft R., Osthus D.: Loose Hamilton cycles in hypergraphs. Discrete Math. 311, 544–559 (2011)
Kierstead H., Sárközy G., Selkow S.: On k-ordered Hamiltonian graphs. J. Graph Theory 32, 17–25 (1999)
Knox F., Kühn D., Osthus D.: Approximate Hamilton decompositions of random graphs. Random Struct. Algorithms 40, 133–149 (2012)
Kostochka A., Yu G.: An extremal problem for H-linked graphs. J. Graph Theory 50, 321–339 (2005)
Kriesell M.: All 4-connected line graphs of claw-free graphs are Hamiltonian connected. J. Combin. Theory B 82, 306–315 (2001)
Krivelevich, M.: On the number of Hamilton cycles in pseudo-random graphs. Electron. J. Combin. 19 (2012), Paper 24
Krivelevich M., Samotij W.: Optimal packings of Hamilton cycles in sparse random graphs. SIAM J. Discrete Math. 26, 964–982 (2012)
Krivelevich M., Sudakov B.: Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory 42, 17–33 (2003)
Kronk H.: A generalization of a theorem of Pósa. Proc. Am. Math. Soc. 21, 77–78 (1969)
Kühn D., Lapinskas J., Osthus D.: Optimal packings of Hamilton cycles in graphs of high minimum degree. Combin. Prob. Comput. 22, 394–416 (2013)
Kühn D., Osthus D.: Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree. J. Combin. Theory Ser. B 96, 767–821 (2006)
Kühn D., Osthus D.: Hamilton decompositions of regular expanders: a proof of Kelly’s conjecture for large tournaments. Adv. Math. 237, 62–146 (2013)
Kühn D., Osthus D.: A survey on Hamilton cycles in directed graphs. Eur. J. Combin. 33, 750–766 (2012)
Kuipers, E.J., Veldman, H.: Recognizing claw-free Hamiltonian graphs with large minimum degree. Memo, 1437, Department of Applied Mathematics, University of Twente, Enschede (1998)
Lai H.-J., Shao Y., Yu G., Zhan M.: Hamiltonian connectedness in 3-connected line graphs. Discrete Appl. Math. 157, 982–990 (2009)
Lai H.-J., Shao Y., Zhan M.: Hamiltonicity in 3-connected claw-free graphs. J. Combin. Theory B 96, 493–504 (2006)
Lai H.-J., Shao Y., Zhan M.: Every 4-connected line graph of a quasi-claw-free graph is Hamiltonian connected. Discrete Math. 308, 5312–5316 (2008)
Lai H.-J., Xiong L., Yan H., Yan J.: Every 3-connected claw-free Z 8-free graph is Hamiltonian. J. Graph Theory 64, 1–11 (2010)
Lafond, M., Seamone, B.: Some Hamiltonian chordal graphs are not cycle extendable (preprint)
Las Vergnas, M.: Thesis, University of Paris, Paris (1972)
Leach C.D., Rodger C.: Hamilton decompositions of complete graphs with 3-factor leaves. Discrete Math. 279, 337–344 (2004)
Leach C.D., Rodger C.: Hamilton decompositions of complete multipartite graphs with any 2-factor leave. J. Graph Theory 44, 208–214 (2003)
Lee C., Sudakov B.: Dirac’s theorem for random graphs. Random Struct. Algorithms 41, 293–305 (2012)
Li G., Lu M., Liu Z.: Hamiltonian cycles in 3-connected claw-free graphs. Discrete Math. 250, 137–151 (2002)
Li R.: Hamiltonicity of 3-connected quasi-claw-free graphs. Discrete Math. 265, 393–399 (2003)
Li R., Schelp R.: Every 3-connected distance claw-free graph is Hamiltonian connected. Discrete Math. 268, 185–197 (2003)
Li M., Chen X., Broersma H.: Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs. J. Graph Theory 68, 285–298 (2011)
Li J., Shen R., Tian F.: Cycles containing given subsets in 1-tough graphs. Ars Combin. 58, 193–204 (2001)
Li M.: Hamiltonian connected claw-free graphs. Graphs Combin. 20, 341–362 (2004)
Lu, L., Székely, L.: Using Lovász local lemma in the space of random injections. Electron. J. Combin. 14, 13 (2007) Paper 63
Luczak T., Pfender F.: Claw-free 3-connected P 11-free graphs are Hamiltonian. J. Graph Theory 47, 111–121 (2004)
Malkevitch, J.: Polytopal graphs. In: Beineke, Wilson (eds.) Selected Topics in Graph Theory, vol. 3. Academic Press, New York, pp. 169–188 (1988)
Manikandan R.S., Paulraja P.: Hamiltonian decompositions of the tensor product of a complete graph and a complete bipartite graph. Ars Combin. 80, 33–44 (2006)
Manikandan R.S., Paulraja P.: Hamilton cycle decompositions of the tensor product of complete multipartite graphs. Discrete Math. 308, 3586–3606 (2008)
Matthews M., Sumner D.: Hamiltonian results in K 1,3-free graphs. J. Graph Theory 8, 139–146 (1984)
Mohar B.: A domain monotonicity theorem for graphs and hamiltonicity. Discrete Appl. Math. 36, 169–177 (1992)
Moon J., Moser L.: On Hamiltonian bipartite graphs. Isr. J. Math. 1, 163–165 (1963)
Müller T., Pérez-Giménez X., Wormald N.: Disjoint Hamilton cycles in the random geometric graph. J. Graph Theory 68, 299–322 (2011)
Nakamoto A., Ozeki K.: Hamiltonian cycles in bipartite quadrangulations on the torus. J. Graph Theory 69, 143–151 (2012)
Nash-Williams, C.St.J.A.: Edge-Disjoint Hamiltonian Circuits in Graphs Having Sufficiently Large Valencies. Studies in Pure Mathematics. Academic Press, London, pp. 157–183 (1971)
Nash-Williams, C.St.J.A.: Unexplored and Semi-Explored Territories in Graph Theory. New Directions in Graph Theory. Academic Press, New York, pp. 169–176 (1973)
Nikoletseas S., Raptopoulos C., Spitakis P.G.: On the independence number and Hamiltonicity of uniform random intersection graphs. Theor. Comput. Sci. 412, 6750–6760 (2011)
Ng L., Schultz M.: k-ordered Hamiltonian graphs. J. Graph Theory 24, 45–57 (1997)
Okol’nishnikova E.A.: On the number of Hamiltonian cycles in dense Hamiltonian graphs. (Russ.) Mat. Tr. 8, 199–206 (2005)
Ore, O.: A note on Hamiltonian circuits. Am. Math. Mon. 67, 55 (1960)
Perez Reilly, E., Scheinerman, E.: Random threshold graphs. Electron. J. Combin. 16, Paper 130 (2009)
Pfender F.: Hamiltonicity and forbidden subgraphs in 4-connected graphs. J. Graph Theory 49, 262–272 (2005)
Pike D.A.: Hamilton decompositions of line graphs of some bipartite graph. Discuss. Math. Graph Theory 25, 303–310 (2005)
Pósa L.: On the circuits of finite graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 355–361 (1963)
Qiao S., Zhang S.: Spanning cyclic subdivisions of vertex-disjoint cycles and chorded cycles in graphs. Graphs Combin. 28, 277–285 (2012)
Robinson R.W., Wormald N.: Almost all regular graphs are Hamiltonian. Random Struct. Algorithms 5, 363–374 (1994)
Rodger C.: Hamiltonian decomposable graphs with specified leaves. Graphs Combin. 20, 541–543 (2004)
Rödl V., Rucinski A., Szemerédi E.: A Dirac-type theorem for 3-uniform hypergraphs. Combin. Probab. Comput. 15, 229–251 (2006)
Rödl V., Rucinski A., Szemerédi E.: An approximate Dirac-type theorem for k-uniform hypergraphs. Combinatorica 28(2), 229–260 (2008)
Rödl V., Ruciński A., Szemerédi E.: Dirac-type conditions for Hamiltonian paths and cycles in 3-uniform hypergraphs. Adv. Math. 227, 1225–1229 (2011)
Rosenfeld M., Barnette D.: Hamiltonian circuits in certain prisms. Discrete Math. 5, 389–394 (1973)
Ryjáček Z.: On a closure concept in claw-free graphs. J. Combin. Theory B 70, 217–224 (1997)
Ryjáček Z., Vrána P.: Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs. J. Graph Theory 66, 152–173 (2011)
Rybarczyk, K.: Sharp threshold functions for random intersection graphs via a coupling method. Electron. J. Combin. 18, Paper 36 (2011)
Sakai T.: Long paths and cycles through specified vertices in k-connected graphs. Ars Combin. 58, 33–65 (2001)
Sanders D.P.: On paths in planar graphs. J. Graph Theory 24, 341–345 (1997)
Sárkozy G., Selkow S., Szemerédi E.: On the number of Hamiltonian cycles in Dirac graphs. Discrete Math. 265, 237–250 (2003)
Sárkozy G., Selkow S.: Distributing vertices along a Hamiltonian cycle in Dirac graphs. Discrete Math. 308, 5757–5770 (2008)
Sciriha I., Cardoso D.M.: Necessary and sufficient conditions for a Hamiltonian graph. J. Combin. Math. Combin. Comput. 80, 127–150 (2012)
Shepherd F.B.: Hamiltonicity in claw-free graphs. J. Combin. Theory B 53, 173–194 (1991)
Skupien Z.: Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamilton cycles. Discrete Math. 309, 6382–6390 (2009)
Sudakov B., Vu V.: Local resilience of graphs. Random Struct. Algorithms 33, 409–433 (2008)
Sugiyama T.: Hamiltonian cycles through a linear forest. Sut. J. Math. 40, 103–109 (2004)
Tait, P.G.: Remark on the coloring of maps. Proc. R. Soc. Edinb. 10, 729 (1880)
Thomas R., Yu X.: 4-connected projective planar graphs are Hamiltonian. J. Combin. Theory B 62, 114–132 (1994)
Thomas R., Yu X.: Five-connected toroidal graphs are Hamiltonian. J. Combin. Theory B 69, 79–96 (1997)
Thomas R., Yu X., Zang W.: Hamilton paths in toroidal graphs. J. Combin. Theory B 94, 214–236 (2005)
Thomassen C.: A theorem on paths in planar graphs. J. Graph Theory 9, 169–176 (1983)
Thomason A.G.: Hamiltonian cycles and uniquely edge colorable graphs. Ann. Discrete Math. 3, 259–268 (1978)
Thomassen C.: Reflections on graph theory. J. Graph Theory 10, 309–324 (1986)
Tian F., Wei B.: Pancyclicity mod k of K 1,4-free graphs. Adv. Math. (China) 34, 221–232 (2005)
Tutte W.: A theorem on planar graphs. Trans. Am. Math. Soc. 82, 309–324 (1956)
Tuza Z.: Steiner system and large non-Hamiltonian hypergraphs. Matematiche 61, 179–183 (2006)
van den Heuvel, J.: Hamilton cycles and eigenvalues of graphs. Linear Algebra Appl. 226–228, 723–730 (1995)
Verrall H.: Hamiltonian decompositions of complete 3-uniform hypergraphs. Discrete Math. 132, 333–348 (1994)
Westland E., Liu J., Kreher D.: 6-regular Cayley graphs on abelian groups of odd order are Hamiltonian decomposable. Discrete Math. 309, 5106–5110 (2009)
Whitney H.: A theorem on graphs. Ann. Math. 32, 378–390 (1931)
Witte D., Gallian J.: A survey: Hamiltonian cycles in Cayley graphs. Discrete Math. 51, 293–304 (1984)
Yang Z.: Note on F-Hamiltonian graphs. Discrete Math. 196, 281–286 (1999)
Yu X.: Disjoint paths, planarizing cycles, and spanning walks. Trans. Am. Math. Soc. 349, 1333–1358 (1997)
Zhou J., Lin C., Hu G.: Spectral radius of Hamiltonian planar graphs and outerplanar graphs. Tsinghua Sci. Technol. 6, 350–354 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gould, R.J. Recent Advances on the Hamiltonian Problem: Survey III. Graphs and Combinatorics 30, 1–46 (2014). https://doi.org/10.1007/s00373-013-1377-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-013-1377-x