Hamiltonian paths and cycles in vertex transitive graphs

Importance: High ✭✭✭
Author(s): Lovasz, Laszlo
Recomm. for undergrads: no
Posted by: mdevos
on: March 18th, 2007

\begin{problem} Does every connected \Def{vertex-transitive graph} have a \Def{Hamiltonian path}? \end{problem}

The question posed here is due to Lovasz [L], but the general problem of finding Hamiltonian paths and cycles in highly symmetric graphs is much older. Knuth has traced it back to \Def[bell ringing]{bell ringing}, and it appears again in \Def{gray codes} and in the \Def{knight's tour} of a chessboard.

Vertex-transitive graphs are, of course, very special, very well-behaved graphs, and it seems unsurprising that many of them have Hamiltonian cycles. What is surprising is that there are only five connected ones known which do not have Hamiltonian cycles. This list consists of the complete graph on 2 vertices, the \Def{Petersen graph}, \href[Coxeter's graph]{http://mathworld.wolfram.com/CoxeterGraph.html}, and the graphs obtained from Petersen and Coxeter by truncating every vertex (inflate each vertex to a triangle). In particular, we do not know of a vertex transitive graph without a Hamiltonian path.

Interestingly, there seems to be considerable disagreement among experts as to what the answer will be. On one hand, there does not appear to be any particular reason why vertex-transitive graphs should almost always have Hamiltonian cycles. On the other hand, such graphs have been studied and searched for at great length, and so far every one investigated with the exception of the five listed above has proved to have a Hamiltonian cycle. Babai formulated the following conjecture which is in quite sharp contrast to the problem above.

\begin{conjecture}(Babai [B96]) There exists $\epsilon > 0$ so that there are infinitely many connected vertex-transitive graphs $G$ with longest cycle of length $<(1-\epsilon)|V(G)|$. \end{conjecture}

For general vertex-transitive graphs, very little is known. Babai [B79] has shown that a vertex-transitive graph on $n$ vertices has a cycle of length $\ge \sqrt{3n}$, but (though a very clever arguement) this is obviously quite far from the conjecture. Considerable attention has been given to the special case of \Def[Cayley graphs]{cayley graph}. Here we have the following conjecture.

\begin{conjecture} Every connected Cayley graph (apart from $K_2$) has a Hamiltonian cycle. \end{conjecture}

The above conjecture is not difficult to prove for abelian groups. Witte [W] proved it for $p$-groups, and it has also been established for certain special types of generating sets. Two other results of note are a theorem of Pak-Radocic [PR] showing that every group $G$ has a generating set of size $\le \log_2(|G|)$ for which the corresponding Cayley graph is Hamiltonian, and a theorem of Krivelevich-Sudakov [KS] showing that almost surely taking a random set of $\log^5(|G|)$ elements of $G$ as generators yields a Hamiltonian graph.

Bibliography

[B79] L. Babai, Long cycles in vertex-transitive graphs. J. Graph Theory 3 (1979), no. 3, 301--304. \MRhref{0542553}

[B96] L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447-1540. \MRhref{1373683}

[KS] M. Krivelevich and B. Sudakov, \href[Sparse pseudo-random graphs are Hamiltonian]{http://www.math.princeton.edu/~bsudakov/pseudo-hamiltonian.pdf}. J. Graph Theory 42 (2003), no. 1, 17--33. \MRhref{1943104}

[L] L. Lov\'{a}sz, "Combinatorial structures and their applications", (Proc. Calgary Internat. Conf., Calgary, Alberta, 1969), pp. 243-246, Problem 11, Gordon and Breach, New York, 1970.

[PR] I. Pak and R. Radocic, \href[Hamiltonian paths in Cayley graphs]{http://www-math.mit.edu/~pak/hamcayley8.pdf}, preprint

[W] D. Witte, Cayley digraphs of prime-power order are Hamiltonian. J. Combin. Theory Ser. B 40 (1986), no. 1, 107--112. \MRhref{0830597}

[WG] D. Witte and J.A. Gallian, A survey: Hamiltonian cycles in Cayley graphs. Discrete Math. 51 (1984), no. 3, 293--304. \MRhref{0762322}


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