On the Number of Hamiltonian Cycles in Tria n g u lat ions

Jan Kratochvil CHARLES UNl VERSI TY

PRAGUE. CZECHOSLOVAKIA

Dainis Zeps P STUTSCHKA UNIVERSITY

RIGA, CZECHOSL 0 VA KIA

ABSTRACT

It is proved that if a planar triangulation different from K3 and K4 contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [21, this yields that, for n 2 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3-connected triangu- lated graphs.

1. PRELIMINARY

In 1978, E. Grinberg [ l ] found examples of 3-connected (nonplanar) graphs with exactly one Harniltonian cycle and he asked whether a planar graph of this property exists. In 198 1. L. S . Melnikov [personal communication] asked a more special question, whether there exists a planar triangulation with exactly one Hamiltonian cycle (different from the trivial example K?) . In the sequel, the negative answer to thc latter question is proved. Moreover. we show that a Hamiltonian planar triangulation on more than four vertices has at least four Hamiltonian cycles. This provides a solution of the main problem asked in [ 2 ] , where it is shown that, for every n 2 12, there exists a planar triangulation on n vertices containing exactly four Harniltonian cycles. We extend our result to triangulations of arbitrary surfaces and to (vertex-)3-connected graphs, and more generally to the so-called two-triangle graphs defined below. The proof is based on two propositions; the idea of thc so-called lollipop graphs 14) is used in the proof of Proposition 2.

Journal of Graph Theory, Vol. 12, No. 2, 191-194 (1988) 0 1988 by John Wiley & Sons, Inc. CCC 0364-9024/88/020191-04$04.00

192 JOURNAL OF GRAPH THEORY

All graphs considered are finite, undirected, and without loops and multiple edges. The vertex-set and the edge-set of a given graph G are denoted by V ( G ) and E ( G ) , respectively, and the edges are considered as two-element subsets of the vertex-set. The set of all triangles (i.e., subgraphs isomorphic to K,) of a graph G is denoted by A(G).

A planar rriungulation is any maximum planar graph with respect to the set of edges on a given set of vertices. Similarly. a triangulation of a given surface is any maximum graph that permits embedding in this surface (in the sense of [5, p. 531). A triangulated graph is a graph that does not contain an induced cycle of length greater than three.

A graph is called a two-triangle graph (shortly, a 2A-graph) i f each of its edges lies in at least two of its triangles. A Hamiltonian graph that contains at most three Hamiltonian cycles is called a 13H-graph.

2. TWO-TRIANGLE GRAPHS

Definition. Let G be a graph and let A C A(G) be a set of triangles of G, with the triangles considered as three-element subsets of V(G) . We construct a graph G(A) as follows:

V(G(A)) = V(G) U A,

E(G(h)) = {{u, 6) 1 u E 6 E A } .

[Then G(A) is a bipartite graph with parts V(G) and A , the degrees of all ver- tices in the part A are three.]

Let G be a graph and A C A(G) a set of n triangles of G. I f C is a Hamilto- nian cycle in G(A), say C = {{w,, a,}, {6,, wz), . . . ,{a,,, w,}}, where w, E V(G), 6, E A , a n d { w , , w , , , } C 6, f o r i = I , 2 , . . . , I Z ( W , ~ . , = w,), then

= {{wi, w2}, {w?, w3}, . . . , {wn, w , } } is a Hamiltonian cycle in G. It may hap- pen, however, that for distinct Hamiltonian cycles C , and C, in G(A), the cycles c, and r2 coincide, but in that case G is rich in Hamiltonian cycles:

Lemma 1. Let C, and C, be distinct Hamiltonian cycles in the graph G(A), such that c, = Cz. Then either G 2 K , or G has at least n Hamiltonian cycles, where n = card V(G).

Proof. The case of G having at most four vertices is trivial. Suppose n 2 5 and let C, = {{w,, 6,},{61, w2}, . . . , {6,,, w,}} and C2 = {{w,, u,}, {u,, w2>, . . . , {u", w,}} . Without loss of generality we may suppose that 6, # v,. One can show by induction that then 6, # u, and {w,, w,,?} E E(G) for all i = 1 ,2 , . . . , n ( w n + , = w,, w,,.? = w2) . Then for every i , C ( i ) = {{wi, w 2 } , { w 2 , w?}, . . . , {w, - , , w,. ,}, { w , + ~ , w,}, {w,, w , ~ ~ } , . . . ,{wn, w,}} is a Hamiltonian cycle in G. I

HAMILTONIAN CYCLES IN TRIANGULATIONS 193

Lemma 2. such that the graph G(A) is Hamiltonian as well.

Let G be a Hamiltonian 2A-graph. Then there exists A C A(G),

Proof. Let -d be a Hamiltonian cycle in G. Put A(e) = {6 I e E 6 E A(G)} for e E E(G) (now the triangles are considered as three-element sets of edges) and define a system % = (A(G). {A(e) I e E c}). Applying Hall's theorem, we see that (e has a system of distinct representatives A = {6(e)I(e E c}. Then C = Ur-(u,u)ETZ { {u , 6 ( e ) } , {u , 6(e ) } } is a Hamiltonian cycle in G(A). I

Now Lemmas I and 2 yield

Proposition 1. graph B on 2n vertices exists such that

Let G be a 1311-2A-graph on n 2 5 vertices. Then a bipartite

( i ) all degrees of vertices of one part are three, and ( i i ) B is a 13H-graph.

3. THE BIPARTITE GRAPH

Proposition 2. part (which we call A ) being odd. Then

Let B be a bipartite graph with degrees of all vertices in one

( i ) every path of length 2 starting in V(B) -A is in an even number of Hamil-

( i i ) every edge of B is in an even number of Hamiltonian cycles of B , and (iii) the total number of Hamiltonian cycles that occur in B is even.

tonian cycles of B ,

Proof. ( i i ) and (iii) follow from ( i ) , which follows from the method of Thomason [4]. I

4. THEOREM

I t is now clear that a bipartite graph with degrees of all vertices in one part being odd cannot be a 13H-graph. Combining Propositions 1 and 2 then yields

Theorem. then G has at least four Hamiltonian cycles.

If G is a Hamiltonian two-triangle graph with at least five vertices,

Since every 3-connected triangulated graph is a 2A-graph, we get

Corollary 1. most three Hamiltonian cycles is the complete graph on four vertices.

The only Hamiltonian 3-connected triangulated graph having at

I t can be shown that every triangulation of an arbitrary (orientable or nonori- entable) surface is a 2A-graph, provided it has at least four vertices. This gives

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Corollary 2. The only Hamiltonian triangulations of arbitrary surfaces hav- ing at most three Hamiltonian cycles are the complete graphs on three and four vertices.

Let us now define the numbers c,,(n), cIcr(n), and cpT(tz) as the minimum numbers of Hamiltonian cycles in Hamiltonian 2A-graphs. 3-connected tri- angulated graphs, and planar triangulations on n vertices. respectively. The main problem of Hakimi, Schmeichel, and Thomassen stated in 121 is to deter- mine cpr(ti) . In 121, examples are constructed showing that ~ , , ~ ( n ) 5 4 for n 1 12. Since these examples are also 3-connected triangulated, it follows that c,,,(n) = 4 for n 2 12. Then we have proved

Corollary 3. For n 2 12, we have c2,(n) = c,,(n) = L ’ , , ~ ( I z ) = 4.

Remark. Although c?,(4) = c,,,(4) = cp,(4) = 3 and c2,(5) = cic,(5) = ~ ~ ( 5 ) = 6, the numbers c.,(n), c3cT(t?), and c J n ) are not equal in general. One can show that cPT(6) = 10, c,,,(6) = 6, and c2,(6) = 4. (The cases of 4 5

n 5 1 1 are considcred in 131 .)

ACKNOWLEDGMENTS

The authors would like to thank C. Thomassen for calling their attention to the main problem in 121.

References

[ 11 E. Grinberg, Three-connected graphs with exactly one Hamiltonian cycle. Republican Foundation of Algorithms and Programmes. Computing centre. P. Stutschka University, Riga, U.S.S.R. (1986) [in Russian].

[2J S.L. Hakimi, E.F. Schmeichel, and C. Thomassen, On the number of Hamiltonian cycles in a maximal planar graph. J . Graph Theon 3 (1979) 365-370.

131 J. Kratochvil and D. Zeps, On Hamiltonian cycles in two-triangle graphs. Acta Univ. Carolin. (to appear).

[4] A. G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs (Advances in Graph Theory, Cambridge Combinatorics Conference, Trinity College. Cambridge, 1977,) Ann. Discrete Math. 3 (1978) 259-268.

151 A. T. White, Graphs, Groups and Surfaces. North-Holland, Amsterdam (1973).