Lovász (1970) conjectured that every connectedvertex-transitive graph is traceable
(Gould, p. 33). This conjecture was subsequently verified for several special
orders and classes. Furthermore, with a few notable exceptions, such graphs were
also shown to be Hamiltonian. Babai proved the
conjecture for graphs with prime order (Bondy and Murty 1976, Lipman 1985, Gould).
Alspach (1979) showed that every connected vertex-transitive graph of order except the Petersen graph
is Hamiltonian. Marušič (1982) showed
that every connected vertex-transitive graph of order , , , and is Hamiltonian, while
Marušič and Parsons (1983) showed that connected vertex-transitive graphs
of order and are traceable (Gould,
p. 33).
Thomassen (1991) conjectured that there are only a finite number of connected nonhamiltonian vertex-transitive graphs, while Babai (1979, 1996) conjectured that there are infinitely many.
Babai, L. Problem 17 in "Unsolved Problems." In Summer Research Workshop in Algebraic Combinatorics. Simon Fraser University,
Jul. 1979.Babai, L. "Automorphism Groups, Isomorphism, Reconstruction."
Ch. 27 in Handbook
of Combinatorics, 2 vols. (Ed. R. L. Graham, M. Grötschel,
M.; and L. Lovász). Cambridge, MA: MIT Press, pp. 1447-1540, 1996.Bondy,
J. A. and Murty, U. S. R. Graph
Theory with Applications. New York: North Holland, 1976.Bryant,
D. and Dean, M. "Vertex-Transitive Graphs that have no Hamilton Decomposition."
25 Aug 2014. http://arxiv.org/abs/1408.5211.Godsil,
C. and Royle, G. "Hamilton Paths and Cycles." C§3.6 in Algebraic
Graph Theory. New York: Springer-Verlag, pp. 45-47, 2001.Gould,
R. J. "Updating the Hamiltonian Problem--A Survey." n.d. http://www.mathcs.emory.edu/~rg/updating.pdf.Kutnar,
K. and Marušič, D. "Hamilton Cycles and Paths in Vertex-Transitive
Graphs-Current Directions." Disc. Math.309, 5491-5500, 2009.Lipman,
M. "Hamiltonian Cycles and Paths in Vertex-Transitive Graphs with Abelian and
Nilpotent Groups." Disc. Math.54, 15-21, 1985.Lovász,
L. Problem 11 in "Combinatorial Structures and Their Applications." In
Proc. Calgary Internat. Conf. Calgary, Alberta, 1969. London: Gordon and Breach,
pp. 243-246, 1970.Marušič, D. "Hamiltonian Paths
in Vertex-Symmetric Graphs of Order ." Disc. Math.42, 227-242, 1982.Marušič,
D. and Parsons, T. D. "Hamiltonian Paths in Vertex-Symmetric Graphs of
Order ." Disc. Math.43, 91-96, 1983.Royle,
G. "Transitive Graphs." http://school.maths.uwa.edu.au/~gordon/trans/.Thomassen,
C. "Tilings of the Torus and the Klein Bottle and Vertex-Transitive Graphs on
a Fixed Surface." Trans. Amer. Math. Soc.323 605-635, 1991.