AN ALGORITHM WHICH GENERATES THE HAMILTONIAN CIRCUITS OF A CUBIC PLANAR MAP W. L. PRICE ABSTRACT The paper describes an algorithm which generates those Hamiltonian circuits of a given cubic planar map which include some chosen edge of the map. A simple extension of the method results in the generation of all the Hamiltonian circuits of the map. The algorithm can readily be implemented on a digital computer, subject to limitations on storage. A general property of the generating function defined by the algorithm leads to a proof th t the total number of Hamiltonian circuits which include any given edge of a cubic planar map is even. Does the structure of the function yield further information concerning the properties of the family of Hamiltonian circuits? 1. Introduction 1.1. Algorithms which determine the family of Hamiltonian circuits in a graph are generally of one of two kinds: either they are based on algebraic methods which attempt to find all the Hamiltonian circuits at once, or they depend on enumerative techniques in which the Hamiltonian circuits are found one at a time. Christofides (1975) describes an algebraic method based on the work of Yau (1967), Danielson (1968) and Dhawan (1969) which involves successive matrix multiplications, starting with the adjacency matrix for the graph. Such methods make considerable demands on computer storage and processing time and become impracticable for graphs which have more than a few tens of vertices. Enumerative methods, such as those of Roberts and Flores (1966) and Selby (1970), consider one simple chain at a time, the chain being continuously extended until either a Hamiltonian circuit is obtained or it becomes apparent that the chain cannot lead to a Hamiltonian circuit, in which case the chain is modified in a systematic way and the search continues. A relatively small amount of computer storage is required for enumerative search. 1.2. The algorithm described in this paper is specific to cubic planar maps, rather than graphs in general, and is of the type which generates all Hamiltonian circuits simultaneously. The procedure involves iterative multiplication of simple expressions rather than matrix operations, and as such it makes the minimum possible demand on computer storage for a method of this type. The primary purpose of this algorithm is not, however, to determine the Hamiltonian circuits of particular cubic planar maps it was devised by the author principally as a means to establish general properties of the family of Hamiltonian circuits in such maps. It will be shown that the properties of the generating function implicit in the algorithm lead to a proof of the well known result, first published by Tutte (1946), that the total number of Hamiltonian circuits which include any given edge of a cubic planar map is always even. The method seems potentially capable of establishing further properties of the family of Hamiltonian circuits in a cubic planar map and of relating these properties to the topology of the map. Received 13 September, 1976; revised 8 March, 1978. [J. LONDON MATH. SOC. (2), 18 (1978), 193-201]
194 W. L. PRICE 1.3. The term CP-map will be used to denote a cubic planar map which is connected and which has no isthmuses or multiple edges. The dual of such a map will be termed a CP-dual. A CP-dual is planar, connected and has no loops or multiple edges. Each face of a CP-dual is triangular. 1.4. To each Hamiltonian circuit (H-circuit) in a CP-map there corresponds a particular two-tree in the CP-dual such that the tree edges are in one-one correspondence with the non-circuit edges in the CP-map. Such a two-tree will be termed an H-tree; it has the property that one, and only one, tree edge is associated with each face of the CP-dual. Fig. 3 shows the family of seven H-circuits in a particular <5P-map, M l5 together with the set of corresponding H-trees in the CP-dual, D {. 2. The generating algorithm 2.1. The algorithm generates directly the subset {H E } of {//}, where {H} denotes the set of all H-circuits of a CP-map, M, and {H E } represents those members of {H} which include an arbitrarily chosen edge, E, of M. The formal specification of the algorithm will follow a preliminary discussion of the underlying principle. 2.2. To each H-circuit in M which includes E there corresponds an H-tree in the CP-dual, D, which excludes the edge E', where ' in D corresponds to E in M. In the context of this algorithm it is convenient to assign a direction to each of the tree edges of the H-tree, the directions being dependent on the choice of E'. The two vertices in D which are connected by E' are the root vertices of the directed twotree, and the tree edges are directed towards the root vertices. In Fig. 1 an H-tree of a particular CP-dual, D lt is drawn so as to show the edge directions appropriate to the choice of g and h as root vertices. It is evident that from each vertex of D it other than the root vertices, there emanates exactly one outwardly directed edge. This property clearly extends to any H-tree by virtue of the properties of the tree structure. 2.3. Assume that M (and hence D) is given and that some edge E of M (and hence E' of D) is specified. Suppose that a sub-set of the edges of D be assigned directions in such a way as to satisfy each of the two conditions which follow. Condition 1. Let the two vertices connected by E' be the root vertices and from each vertex of D other than the root vertices let one, and only one, of the incident edges be chosen as an outwardly directed edge. If Condition 1 is satisfied then, starting from any vertex other than a root vertex, and following only directed edges, one proceeds by a unique path which either terminates in a root vertex or enters a directed circuit. (The chosen edges cannot form a circuit other than a directed circuit, for this would imply that at least one vertex possesses more than one outwardly directed edge). Condition 2. Let the directed edges be so chosen that not more than one directed edge bounds each triangular face of D. The next step is to prove that if Condition 2 is satisfied then directed circuits cannot arise. Consider a circuit formed by a set of directed edges satisfying Condition 1. Because the two root vertices are connected by the edge ', which is not
THE HAMILTONIAN CIRCUITS OF A CUBIC PLANAR MAP 195 a directed edge and hence is not on the circuit, it follows that the root vertices both lie on the same side of the circuit. Consider that region, R, of D which is bounded by the circuit and which does not contain the root vertices. Let n be the number of edges on the circuit, and let v,f, e be respectively the number of vertices, faces and edges within R. Applying Euler's relation to the region R gives and because each face is triangular it follows that Hence, eliminating e, 3f=n + 2e. 2v+n=f+2. Now the number of directed edges arising within R is v (one for each internal vertex) and each of these is associated with two faces. The number of directed edges on the circuit is n and each of these is associated with one face within R. But 2v + n =/+2 >/, hence it cannot be true that each of the faces is bounded by only one directed edge and Condition 2 is violated. Therefore if both Conditions 1 and 2 are satisfied it follows that circuits cannot arise and the directed edges constitute an H-tree in D to which there corresponds an H-circuit in M within the set 2.4. The foregoing principle suggests the following algorithmic procedure. 1. Let each triangular face of D be arbitrarily assigned an index number (represented by p, q,r...) and each vertex an index letter (represented by a, b,c,... k,...). Let each edge be denoted by an un-ordered number pair (p, q) where p and q are the index numbers of the faces which are separated by that edge. 2. For each vertex, k, other than the two root vertices (those associated with the specified edge E'), form the vertex sum where each term corresponds to an edge incident on that vertex. 3. Form the product function f E. = s a *s b *s c... of all these vertex sums where the product operator * is defined as follows: The product s a * s b is the sum of terms such as (p, q) (r, s) where (p, q) is any term in s a and (r, s) any term in s b such that the index numbers p, q, r, s are all different. Thus, if s a includes the term (1, 2) and s b includes the terms (2, 5) and (3, 4), the product s a * s b will include the term (1, 2) (3, 4) but not (1, 2) (2, 5). No significance attaches to the ordering of the product terms, thus (1, 2) (3, 4) = (3,4) (1, 2) = (4, 3) (1,2) etc. Similarly the product s a * s b * s c is the sum of terms such as (p, q) (r, s) (t, u), formed by taking one term from each of s a, s b and s c, such that p, q, r, s, t, u are all different. The extension to higher order products is obvious. The associative law holds for the product operation so that the vertex sums may be multiplied successively in any order and/. is unique. If/ - # 0 each term in/. is formed by taking one
196 W. L. PRICE component (representing an edge) from each of the vertex sums, thus satisfying Condition 1. Furthermore the definition of the product operation ensures that, in each term of f E., no face index number appears more than once. Hence each face is associated with one, and only one, directed edge and Condition 2 is satisfied. It follows that the set of edges represented by the components of any term of/. form an H-tree in D such that E' is not a tree edge. Conversely, every H-tree having E' as a non-tree edge is uniquely described by a set of directed edges satisfying Conditions 1 and 2, and is therefore expressible as a term of/.. Thus/ E. represents the whole family of such H-trees and hence the algorithm generates the set {H E } in M. 2.5. Although the algorithm has been described in terms of the CP-dual, D, the generating function f E can be obtained directly from the CP-map, M, and the terms of f E interpreted as the H-circuits in M which include the edge E. The components of each term of f E now represent non-circuit edges i.e. if these are deleted from M the remaining edges form a Hamiltonian circuit through E. The procedure will be illustrated by means of a specific example. The CP-map, M ls is shown, together with its corresponding CP-dual, D u in Fig. 2. Vertices in M x (corresponding to faces in Dj) are arbitrarily assigned index numbers and faces in M x are arbitrarily assigned index letters. Suppose that the aim is to form the set {H E } where E is the edge linking vertices 1 and 12 in M x. Observe that E separates the faces g and h which correspond to root vertices in D x. First form the face sums of faces a, b, c, d, e, f. Thus * = [0,2) + (2, 3) + (3,4) + (4, 5) + (5, 1)], etc. Then s a *s b = [(1, 2)(3, 10) +(1, 2)(10, 11) + (2,3)(10, 11) + (4, 5)(2, 3)+ (4, 5)(3, 10)+ (4, 5)(10, 11)+ (4, 5)(11, 2) + (5, 1)(2, 3) + (5, 1)(3, 10) + (5, l)(10, ll) + (5, 1)(11, 2)]. The subsequent steps in forming the product function are straightforward and one finally obtains IE = s a * s b * s c * s d * s e * s f (2,3)(1,5)(8, 10)(4,9)(6, 7)(11,12) The first term in f E represents the H-circuit obtained by deleting the edges which join vertices 1 and 2, 3 and 10, 4 and 5, etc. in M x This circuit is shown in Fig. 3(e). Similarly the other terms of f E represent the other three members of {H E } shown in Figs. 3(g), 3(f) and 3(d). The operations involved in the determination of f E are readily expressible in the form of a computer programme. Hence, subject to the limitations of storage, the algorithm can be implemented on a digital computer. 12)].
THE HAMILTONIAN CIRCUITS OF A CUBIC PLANAR MAP 197 2.6. The procedure is modified if the aim is to obtain {H}, the entire family of H-circuits. Clearly every member of {H} passes through two of the three edges incident at any vertex. Choose, any vertex, vertex 1 for example, and form the partial product function P = s b * s c * s d * s e * s f which excludes the terms s a, s g, s h corresponding to the three faces incident at vertex 1. Then P * s a generates the H-circuits which pass through edge E, separating g and //, as previously shown. Similarly P * s g generates the H-circuits which pass through the edge which separates a and h, and P s h generates the H-circuits which pass through the edge which separates a and g. Thus the function (s a +s g +s h )* P generates (twice) each member of {H}. Application of this procedure to M x results in the family of seven H-circuits shown in Fig. 3. 3. Properties of the generating function 3.1. The generating function, f E, and its partial products possess a simple parity property. Consider the face sum, s, for any face of M. Let a k be the number of terms in s which contain the vertex number k and a kl be the number of terms which contain both k and /. It is evident from the definition of s that a k = 2 or 0 according as k is, or is not, a vertex on the boundary of the face which generates s. The value of a kl is 1 if A: and / are joined by an edge which bounds the face otherwise, a kl = 0. Let CJE be the number of terms in s which do not contain the index number k, and 0-fj be the number of terms which contain neither k nor /. Then o- E = S~a k and a m = S-a k -a l + a kl where S is the total number of terms in.y. Consider next a product t of any number of face sums Let t = s a *s b *s c... be defined for / in the same way as are defined for s. Now suppose that a new partial product t' is obtained by forming the product of t with some face sum s, giving t' = t * s. From the definition of the product operator, it follows that the number of terms in t' which contain some arbitrary index number p is given by where the summation ranges over all index numbers k except k = p. Hence V = I k where T is the total number of terms in t.
198 W. L. PRICE But therefore k V = a p r + T p S-2<yr p + 22 u pk x pk k Now, o k is even for all k. If it is assumed that x k is even for all k it follows that x p ' is even whatever the parity of a pk and x pk. Because p was chosen arbitrarily x k ' is even for all k. Hence, by induction, x k is even for any partial product and, in particular, for the complete generating function f E. An extension of the argument shows that, for any partial product, T fc> x klm> X klmnoi e * C - are all even (where x klm is the number of terms which contain all of A", / and m etc.) whereas may be either odd or even. 3.2. It has been shown that the number of terms in the generating function f E which contain any given index number, k, is even. But each term of f E contains every index number, hence f E always has an even number of terms. Therefore the total number of H-circuits which pass through any given edge of a CP-map is even. 3.3. The properties of the generating function, and its partial products, derived above depend only upon a simple property of the expression for the face sum and on the nature of the product operation. By taking into account the additional combinatorial constraints implied by the topological structure of a CP-map it seems possible that further properties of f E could be found. What are these properties? Do they yield the necessary and sufficient conditions that/ = 0 for some edge i.e. that the edge E is not contained in any H-circuit? What are the necessary and sufficient conditions that/ = 0 for every edge i.e. the CP-map is non-hamiltonian? Fig. 1. Directed edges of an H tree in
THE HAMILTONIAN CIRCUITS OF A CUBIC PLANAR MAP 199 E' (b) D, Fig. 2. Generation algorithm for M x
200 W. L. PRICE H-Circuits in A/ t Fig. 3. H-Trees in Di
THE HAMILTONIAN CIRCUITS OF A CUBIC PLANAR MAP 201 References 1. N. Christofides, Graph Theory An Algorithmic Approach (Academic Press, London, 1975). 2. G. H. Danielson, " On finding the simple paths and circuits in a graph ", IEEE Trans., CT-15 (1968), 294. 3. V. Dhawan, Hamiltonian circuits and related problems in graph theory, M.Sc. Report (Imperial College, London, 1969). 4. S. M. Roberts and B. Flores, " Systematic generation of Hamiltonian circuits ", Com. of ACM, 9 (1966), 690-694. 5. G. R. Selby, The use of topological methods in computer-aided circuit layout, Ph.D. Thesis (London University, 1970). 6. W. T. Tutte, "On Hamiltonian circuits", J. London Math. Soc, 21 (1946), 98-101. 7. S. S. Yau, " Generation of all Hamiltonian circuits, paths and centres of a graph and related problems ", IEEE Trans., CT-14 (1967), 79-81. Engineering Department, University of Leicester, LEI 7RH.