The World of Hamiltonian Circuits
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Hamiltonian circuits have an astonishing variety of settings where they can be applied in problems outside of mathematics (e.g. traveling salesman inspired questions) as well as raising a variety of interesting aspects for many problems in graph theory, combinatorics, and geometry.
Certain classes of important graphs are easily shown to be hamiltonian. For example, the d-cubes, for d at least 2 have hamiltonian circuits.
1. How many inequivalent hamiltonian circuits are there on a d-cube?
Here there can be different definitions of "inequivalent." One natural definition is to consider to HC's different if they use different edges. However, another definition is to consider to HC's different if one can not be transformed into the other by a symmetry of the cube (e.g. an automorphism of the graph of the cube).
The reason for the interest in this topic is related to the construction of Gray Codes for the binary sequences of length d and for ideas related to error-correction technologies.
Richard Hamming showed that how to construct error correcting codes whose code words are strings of binary digits, which can be interpreted as vertices of a d-dimensional cube. The dramatic insight here is that if two strings of the same length from an alphabet differ in k positions then one can think of these strings as being distance k apart. This metric is now known as the Hamming distance in honor of Richard Hamming. If one has a collection of code words, with the property that the Hamming Distance between any pair of code words is at least 2d +1, then the code based on these code words can be used to correct up to d errors per code word. This is done by decoding any received string (which may or may not be a code word) to the closest code word to the received string as measured by the Hamming Distance. It is interesting to note the rapid falloff in the largest number of code words that can be present in an error correcting code for a fixed length n binary sequence as the number of errors increases.
2. Some plane 3-valent graphs which are 3-connected have the property that there are:
a. Edges which belong to every hamiltonian circuit of the graph
b. Edges which belong to no hamiltonian circuit of the graph
It is of interest to investigate which kinds of 3-polytopal graphs have edges of this kind and how many edges of these kinds can occur.
It is also of interest to study 3-valent 3-polytopal graphs with the property that every lies on precisely h hamiltonian circuits of the graph. For example, the graph of the tetrahedron T has the property that every edge lies on exactly 3 hamiltonian graphs. If P arises from T by a sequence of vertex truncations, it is not difficult to see that every edge of P lies on exactly 3 hamiltonian circuits. Thus, we have found an infinite family of convex polyhedra in 3-space for which exactly 3 HC's visit every edge.
Another circle of ideas opened up via the theory of hamiltonian circuits is that of a Gray Code. The "classical Gray Code" involves ordering the 2d binary sequences (e.g. sequences of zeros and ones) into a cyclic sequence so that two consecutive binary sequences in cyclic list differ in only one position. Such a cyclic sequence correspond to the vertices of a d-cube arranged in the order generated from a hamiltonian circuit.
(Gray codes are named for Frank Gray who got his doctorate from the University of Wisconsin in 1916 and spent many years at Bell Telephone Laboratories. His idea is incorporated in Patent 2,632,058 (March 17, 1953).)
Here is another example in this spirit. Suppose one has a graph G and Ti and Tj are two spanning trees of G, one can computed the "distance" between these two trees as follows:
What is the minimum number of operations of the following kind that is necessary to transform one of the trees into the other:
Given a tree T, take some edge of the graph e, which is not in T and add it to the edges of T to obtain a unique circuit C. Now delete some edge of C other than e to get a new tree T'.
We can form a graph H whose vertices are the spanning trees of G and join two vertices of H together with an edge when the spanning trees they represent can be transformed into one another with exactly one of the operations of the type described above. We are now interested in knowing whether or not the graph H has a hamiltonian path or a hamiltonian circuit, which would provide a listing of the spanning trees of G ordered in a way that that to go from one of these spanning trees to another only involves a small change.
Work has been done on constructing "gray codes" for many classes of combinatorial objects. These include the subsets of size k selected from a set of size n, and the partitions of a positive integer n.
There are many open questions about hamiltonian circuits on higher dimensional polytopes. An easy to state conjecture of David Barnette is still open:
Does every 4-valent 4-polytope have a hamiltonian circuit?
It is know that d-polytopal graphs are d-connected (a theorem of Michel Balinksi) but such graphs can not be planar for d greater than 3. Four-connected planar graphs are known to be hamiltonian by a theorem of Tutte. The major difficulty with d-connected graphs (d large, or even 4) is that it is not easy to determine if the graph is that of a d-polytope or not.
Another interesting class of questions arises from finding hamiltonian circuits on prisms formed from graphs. Given two copies of the same graph one can form a prism from them by joining corresponding vertices of the two graphs with edges. A graph H may not have a hamiltonian path (for example, K4,2) but a prism formed from two copies of this graph (Figure 1) will have a hamiltonian circuit. These problems are related to questions about spanning trees of the polytopes. For 3-polytopes David Barnette showed that every such polytope has a spanning tree of maximal valence 3. However, for 4-polytopes there is no uniform value of k which will guarantee that a 4-polytope has a spanning tree of maximal valence k, a dramatic difference between the behavior of 3-polytopes and 4-polytopes.
Wilf, H. Combinatorial Algorithms: An Update, SIAM, Philadelphia, 1989.
Ryjáãek, Z. and T. Kaiser, D. Kráº, M. Rosenfeld, H.-J., Hamiltonian cycles in prisms over graphs, Journal of Graph Theory (to appear).