Contractible Hamiltonian Cycles in Triangulated Surfaces

A map is called {p, q} equivelar if each vertex is incident with exactly q numbers of p-gons. If p = 3 then the map is called a q -equivelar triangulation or a degree -regular triangulation of type q. A map is called a Simplicial Complex if each of its faces is a simplex. Thus a triangulation is a Simplicial Complex For a simplicial complex K, the graph consisting of the edges and vertices of K is called the edge-graph of K and is denoted by EG(K).

If X and Y are two simplicial complexes, then a (simplicial) isomorphism from X to Y is a bijection φ : V (X) → V (Y ) such that for σ ⊂ V (X), σ is a simplex of X if and only if φ(σ) is a simplex of Y . Two simplicial complexes X and Y are called (simplicially) isomorphic (and is denoted by X ∼ = Y ) when such an isomorphism exists. We identify two complexes if they are isomorphic. An isomorphism from a simplicial complex X to itself is called an automorphism of X. All the automorphisms of X form a group, which is denoted by Aut(X).

In 1956, Tutte [13] showed that every 4-connected planar graph has a Hamiltonian cycle. Later in 1970, Grünbaum conjectured that every 4-connected graph which admits an embedding in the torus has a Hamiltonian cycle. In the same article he also remarked that -probably there is a function c(k) such that each c(k)-connected graph of genus at most k is Hamiltonian.

In 1972, Duke [8] showed the existence of such a function and gave an estimate

where k ≥ 1. A. Altshuler [1], [2] studied Hamiltonian cycles and paths in the edge graphs of equivelar maps on the torus. That is in the maps which are equivelar of types {3, 6} and {4, 4}. He showed that in the graph consisting of vertices and edges of equivelar maps of above type there exists a Hamiltonian cycle. He also showed that a Hamiltonian cycle exists in every 6-connected graph on the torus.

In 1998, Barnette [5] showed that any 3-connected graph other than K 4 or K 5 contains a contractible cycle or contains a simple configuration as subgraphs.

In this article we present a necessary and sufficient condition for existence of a contractible Hamiltonian cycle in edge graph of an equivelar triangulation of surfaces.

We moreover show that the contractible Hamiltonian cycle bounds a triangulated 2-disk. If the equivelar triangulation of a surface is on n vertices then this disk has exactly n -2 triangles and all of its n vertices lie on the boundary cycle. We begin with some definitions.

If a surface S has an equivelar triangulation on n vertices then the proof of the Theorem 1 is given by considering a tree with n -2 vertices in the dual map of the degree-regular triangulation of the surface. We define this tree as follows :

Definition 4 Let M denote a map on a surface S, which is the dual map of a n vertex degree-regular triangulation K of the surface. Let T denote a tree on n -2 vertices on M. We say that T is a proper tree if :

1. whenever two vertices u 1 and u 2 of T belong to a face F in M, a path

joining u 1 and u 2 in boundary of F belongs to T .

2. any path P in T which lies in a face F of M is of length at most q -2, where M is a map of type {q, 3}.

If v is a vertex of a simplicial complex X, then the number of edges containing v is called the degree of v and is denoted by deg X (v) (or deg(v)). If the number of i-simplices of a simplicial complex X is f i (X) (0 ≤ i ≤ 2), then the number χ(X) = f 0 (X)f 1 (X) + f 2 (X) is called the Euler characteristic of X. A simplicial complex is called neighbourly if each pair of its vertices form an edge.

Non-Orientable degree-regular combinatorial 2-manifold of χ = -2.

this violates the definition of T . Thus u is incident with exactly one face F 3 of M such that F 3 ∈ G. Since, u is an arbitrary end point this hypothesis holds for all the end vertices. If it happens that for some end vertices u 1 and u 2 of T , the corresponding faces W 1 = W 2 ∈ G then we would have u 1 and u 2 on the same face of M but no path on W 1 joining u 1 and u 2 lies in T . This contradicts the definition of T . Thus G has exactly e distinct elements. This proves the lemma. ✷ Lemma 4.4 Let K be a n vertex degree regular triangulation of a surface S. Let M denote the dual polyhedron corresponding to K and T be a n -2 vertex proper tree in M. Let D denote the subcomplex of K which is dual of T . Then D is a triangulated 2-disk and bd(D) is a Hamiltonian cycle in K.

Proof : By definition of a dual, D consists of n -2 triangles corresponding to n -2 vertices of T . Two triangles in D have an edge in common if the corresponding vertices are adjacent in T . It is easy to see that D is a collapsible simplicia

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