Global cycle properties in graphs with large minimum clustering coefficient

References

• S.A. van Aardt, M. Frick, O.R. Oellermann, and J. de Wet, Global cycle properties of locally connected, locally traceable and locally hamiltonian graphs, Discrete Appl. Math. 205 (2016), 171–179. doi: 10.1016/j.dam.2015.09.022
   Web of Science ®Google Scholar
• S.A. van Aardt and J. de Wet, Hamiltonicity and traceability of locally hamiltonian and locally traceable graphs, Private communication.
   Google Scholar
• A. Abueida and R. Sritharan, Cycle extendability and hamiltonian cycles in chordal graphs classes, SIAM J. Discrete Math. 20 (2006), 669–681. doi: 10.1137/S0895480104441267
   Web of Science ®Google Scholar
• J. Akiyama, T. Nishizeki and N. Saito, NP-completeness of the hamiltonian cycle problem for bipartite graphs, J. Inf. Process. 3 (1980), 73–76.
   Google Scholar
• A. Asratian, Some properties of graphs with local Ore condition, ARS Combinatoria 41 (1995), 97–106.
   Web of Science ®Google Scholar
• J.A. Bondy, Pancyclic graphs I, J. Combin. Theory 11 (1971), 80–84. doi: 10.1016/0095-8956(71)90016-5
   Google Scholar
• J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, Berlin, 2008.
   Google Scholar
• A. Borchert, S. Nicol and O.R. Oellermann, Global cycle properties in locally isometric graphs, Discrete Appl. Math. 205 (2016), 16–26. doi: 10.1016/j.dam.2016.01.026
   Web of Science ®Google Scholar
• G. Chartrand, R. Gould and A.D. Polimeni, A note on locally connected and hamiltonian-connected graphs, Israel J. Math. 33 (1979), 5–8. doi: 10.1007/BF02760528
   Web of Science ®Google Scholar
• G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs, fifth edition, Chapman and Hall/CRC Press, Boca Raton, 2011.
   Google Scholar
• G. Chartrand and R.E. Pippert, Locally connected graphs, Časopis pro Pěstovàní Matematiky 99 (1974), 158–163.
   Google Scholar
• G. Chen, R.J. Faudree, R.J. Gould and M.S. Jacobson, Cycle extendability of hamiltonian interval graphs, SIAM J. Discrete Math. 20 (2006), 682–689. doi: 10.1137/S0895480104441450
   Web of Science ®Google Scholar
• L. Clark, Hamiltonian properties of connected locally connected graphs, Congr. Numer. 32 (1981), 199–204.
   Google Scholar
• G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 69–81. doi: 10.1112/plms/s3-2.1.69
   Google Scholar
• V.S. Gordon, Y.L. Orlovich, C. Potts and V.A. Strusevich, Hamiltonian properties of locally connected graphs with bounded vertex degree, Discrete Appl. Math. 159 (2011), 1759–1774. doi: 10.1016/j.dam.2010.10.005
   Web of Science ®Google Scholar
• A.S. Hasratian and N.K. Khachatrian, Some localization theorems on hamiltonian circuits, J. Combin. Theory Ser B 49 (1990), 287–294. doi: 10.1016/0095-8956(90)90032-U
   Web of Science ®Google Scholar
• G.R.T. Hendry, Extending cycles in graphs, Discrete Math. 85 (1990), 59–72. doi: 10.1016/0012-365X(90)90163-C
   Web of Science ®Google Scholar
• G.R.T. Hendry, A strengthening of Kikust’s theorem, J. Graph Theory 13 (1989), 257–260. doi: 10.1002/jgt.3190130212
   Web of Science ®Google Scholar
• W. Imrich and S. Klavˇzar, Product Graphs: Structure and Recognition, Wile-Interscience, New York, 2000.
   Google Scholar
• P.B. Kikust, The existence of a hamiltonian cycle in a regular graph of degree 5 (Russian, Latvian summary), Latvian Mathematical Yearbook 16 (1975), 33–38.
   Google Scholar
• M. Lafond and B. Seamone, Hamiltonian chordal graphs are not cycle extendable, SIAM J. Discrete Math. 29 (2015), 877–887.
   Google Scholar
• M. Li, D.G. Corneil and E. Mendelsohn, Pancyclicity and NP-completeness in planar graphs, Discrete Appl. Math. 98(3) (2000), 219–225. doi: 10.1016/S0166-218X(99)00163-8
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• C.M. Pareek and Z. Skupień, On the smallest non-hamiltonian locally hamiltonian graph, J. Univ. Kuwait (Sci.) 10 (1983), 9–16.
   Google Scholar
• C. Picouleau, Complexity of the hamiltonian cycle in regular graph problem, Theoret. Comp. Sci. 131 (1994), 463–473. doi: 10.1016/0304-3975(94)90185-6
   Web of Science ®Google Scholar
• J. Ugander, B. Karrer, L. Backstrom and C. Marlow, The anatomy of the Facebook social graph, arXiv 1111.4503[cs.Sl].
   Google Scholar
• D.J. Watts and S. Strogatz, Collective dynamics of ‘small world’ networks, Nature 393 (1998), 440–442. doi: 10.1038/30918
   PubMed Web of Science ®Google Scholar
• D.B. West, Research problems, Discrete Math. 272 (2003), 301–306. doi: 10.1016/S0012-365X(03)00207-3
   Google Scholar