On the Number of Hamiltonian Cycles in a Maximal Planar

S. L. Hakimi Graph

NORTHWESTERN UNIVERSITY

E. F. Schmeichel

C. Thomassen

CALIFORNIA STATE UNIVERSITY-SAN JOSE

AARHAUS UNlVERSlrY

ABSTRACT

We consider the problem of the minimum number of Hamiltonian cycles that could be present in a Hamiltonian maximal planar graph on p vertices. In particular, we construct a p-vertex maximal planar graph containing exactly four Hamiltonian cycles for every p L 12. We also prove that every 4-connected maximal planar graph on p vertices contains at least p/(log, p ) Hamiltonian cycles.

1. INTRODUCTION

In this paper we consider only finite undirected graphs without loops or multiple edges. Our terminology and notation will be standard except as indicated.

The question of when a maximal planar graph is Hamiltonian is of considerable interest. In particular, it is known (see [l]) that not every maximal planar graph contains a Hamiltonian cycle. The purpose of this note is to consider the following related question:

Main Problem. Let G be a Hamiltonian maximal planar graph on p vertices. What is the minimum number of Hamiltonian cycles that G could have, in terms of p?

Journal of Graph Theory, Vol. 3 (1979) 365-370 @ 1979 by John Wiley & Sons, Inc. 0364-9042/79/000%0365$01 .OO

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FIGURE 1. Maximal planar graph

This problem was first raised in Ref. 2. It was originally conjectured that G must have at least 4p-14 Hamiltonian cycles if p 2 5 , with equality occurring only for the maximal planar graph in Figure 1. However, we will here construct a p-vertex maximal planar graph with exactly four Hamiltonian cycles, for every p z 1 2 . The authors have not yet determined if there exists an infinite family of Hamiltonian maximal planar graphs each containing fewer than four Hamiltonian cycles.

Noting that the graphs we construct are all 3-connected, we then consider the main question under the additional assumption that the graph is 4-connected. We prove that every 4-connected maximal planar graph on p vertices contains at least p/(log,p) Hamiltonian cycles, and conclude by conjecturing a larger lower bound.

2. MAIN RESULTS

We begin with two easy lemmas; the verifications are left to the reader.

Lemma 1. Let GI be the maximal planar graph shown in Figure 2. Then:

(a) There is no Hamiltonian path in G1-vi from ui to uk, for i = l , 2 , 3 , and j , k f i .

(b) There are exactly two Hamiltonian paths in G, from u1 to v2. Lemma 2. Let G2 be the maximal planar graph shown in Figure 3, with n 2 1. Then:

(a) There is no Hamiltonian path from v1 to v3 (respectively, from u2 to v3) in G2 - u2(respectively, in G, - vJ.

NUMBER OF HAMILTONIAN CYCLES 367

FIGURE 2. Maximal planar graph G,.

(b) There are exactly two Hamiltonian paths in G2 - v3 from u1 to u2.

We now give the following result:

Theorem 1. Let ~ 2 1 2 . Form a p-vertex maximal planar graph G as follows: Begin with the graph G, of Figure 3 (with n = p - ll), and in the interior of the 3-cycle T = (vl, v2, v3 , u,) place the graph G, of Figure 2 so that G, and G2 have precisely the 3-cycle T in common. Then G has exactly four Hamiltonian cycles.

Proof. Suppose at first that G is Hamiltonian, and let C be a Hamiltonian cycle in G. Since C could enter and leave the interior of T only once, it follows that the edges of C that are also edges of G1 must form a Hamiltonian path in G1 or in G1--vk, for some k. In fact, by Lemma l(a) these edges must form a Hamiltonian path in G1 itself from vi to vj, for some i, j .

FIGURE 3. Maximal planar graph G,.

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Clearly neither vi nor vi could be v3, since otherwise the edges of C that are also edges of G2 would be a Hamiltonian path from v, to vj in G , - uk, violating Lemma 2(a). It follows therefore that any Hamiltonian cycle C in G would have to consist of a Hamiltonian path in G , from v1 to v,, together with a Hamiltonian path in G2-03 from v1 to u2.

Since there are exactly two Hamiltonian paths of each of these types by Lemmas l(b) and 2(b), it follows that G is indeed Hamiltonian and contains exactly four Hamiltonian cycles.

We observe that the maximal planar graphs constructed in Theorem 1 all have connectivity three. It seems natural therefore to inquire about the minimum number of Hamiltonian cycles in a 4-connected, maximal planar graph on p vertices. I t is known that every such graph has at least one Hamiltonian cycle by a theorem of Whitney [4], Actually, Whitney proved the following more general result [4, p. 3801:

Lemma 3. Let G be a 4-connected maximal planar graph. Consider a cycle C in G together with the vertices and edges on one side of C, which we shall call the outside. Let x, y be any two vertices of C dividing C into two paths P, and P2 each of which contains both x and y. Suppose

(1) No two vertices of PI are joined by an edge which lies outside of C;and

(2) Either no two vertices of P2 are joined by an edge which lies outside of C, or else there is a vertex z (distinct from x and y) dividing P2 into two paths P3 and P4 each of which contains z such that no pair of vertices in P3 and no pair of vertices in P4 are joined by an edge which lies outside of C.

Then there is a path from x to y using only edges on and outside of C which passes through every vertex on and outside of C exactly once.

I

We can now give the following result:

Theorem 2. Let G be a 4-connected maximal planar graph on p - vertices. Then G contains at least p/(log,p) Hamiltonian cycles.

Proof. Let (v, w) be any edge in G. Then (v, w) will be incident to two facial 3-cycles in G, say vwx and vwy.

We begin by showing that G contains a Hamiltonian cycle containing the path xvwy. Let the vertices incident to v be w, x, v,, v2,. . . ,urn, y, and let the vertices incident to w be v, y, wl, w2,. . . , w,, x (see Fig. 4). Consider the cycle C = xv1v2. * * vmyw1w2~ - w,x in G. The two vertices x, y partition this cycle C into two paths satisfying the conditions of Lemma 3. [For if there were an edge of the form say (vi, vi) outside of C, then viviv would be a separating 3-cycle in G, violating the fact

NUMBER OF HAMILTONIAN CYCLES 369

V

Vm Y WI

FIGURE 4. An illustration used in the proof

that G is 4-connected.] But then the path from x to y outside of C described in Lemma 3, together with the path xvwy, would be the desired Hamiltonian cycle in G.

For each edge (v, w ) in G, select a Hamiltonian cycle containing the path xvwy. This gives 3 p - 6 (not necessarily distinct) Hamiltonian cycles in G. Let a be the largest number of times that any one Hamiltonian cycle occurs in this way. If we let h(G) denote the number of distinct Hamiltonian cycles in G, it follows that h(G) 2 (3p - 6)Ia.

Consider any Hamiltonian cycle H that is counted a times in this way. Then at least a/3 of the corresponding 4-cycles vxwyv will bound regions which will pairwise intersect in at most one vertex (see Fig. 5 ) . For each of these ( 4 3 ) 4-cycles vxwyv, we can obtain a new Hamiltonian cycle by replacing the path xvwy with the path xwvy. We conclude therefore, that h(G) 2 2a’3. Hence log2h(G) 2 a/3 2 ( p - 2)/h(G), or h(G) log, h(G) z p -2. It follows easily from this that h(G)> p/10g2 p as asserted.

The lower bound given in Theorem 2 appears far from tight. In fact, we offer the following:

Conjecture. If G is a 4-connected maximal planar graph on p vertices, then G contains at least 2 ( p - 2 ) ( p - 4) Hamiltonian cycles, with equality if and only if G is the graph Kz+Cp-z of Figure 6.

I

V

FIGURE 5. Hamiltonian cycle H.

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FIGURE 6. Graph <2+Ccp--2.

Finally, it was noted earlier that not every maximal planar graph is Hamiltonian. Actually, the length of the longest cycle in a p-vertex maximal planar graph may be arbitrarily small compared to p. In particular, if l (G) denotes the length of the longest cycle in a graph G, and if l(p) =min Z(G), the minimum being taken over all p-vertex maximal planar graphs G, then it is known (see [3]) that

Consequently, we have that

lim [l(p)/p] = 0 and also that lim l(p) = 00. P-- P-

ACKNOWLEDGMENTS

The authors wish to express their gratitude to J. A. Bondy for helpful suggestions. This work was supported by the U.S. Air Force Office of Scientific Research under Grant No. AFOSR-76-3017 and the National Science Foundation under Grant No. ENG79-09724.

References

[l] D. Barnette and E. Jucovic, Hamiltonian circuits on 3-polytopes. J. Combinatorial Theory Ser. B 9 (1970) 54-59.

[2] S. L. Hakimi and E. F. Schmeichel, On the number of cycles of length k in a maximal planar graph. J. Graph Theory 3 (1979) 69-86.

[3] J. W. Moon and L. Moser, Simple paths on polyhedra. Pac. J. Math.

[4] H. Whitney, A theorem on graphs. Ann. Math. 32 (1931) 378-390. 13 (1963) 629-31.