The Smallest Nontoroidal Chiral Maps

Steve Wilson MlCHlGAN STATE UNlVERSlN

ABSTRACT In this paper, we display those chiral maps which have fewest edges among those of genus greater than one. One of these is a branched covering of a regular map on the torus, and the other is its dual.

A m a p is a division of a compact 2-manifold into simply connected regions, called the faces of the map, by an embedded graph or multi- graph. The map is called rotary provided that for some face F and vertex V incident with F, there are automorphisms R and S of the map which act as rotations one step around F and V, respectively. A rotary map is regular provided that it also possesses an automorphism X that acts as a reflec- tion about the diameter of face F that contains V. A map that is rotary but not regular is called chiral. (Nomenclature differs: many authors would use the terms “regular,” “reflexible,” and “irreflexible,’y for “rot- ary,” “regular,” “chiral,” respectively.) A nonorientable rotary map is regular [3, p. 1021. The group of automorphisms of a rotary map is transitive in edges, faces, and vertices. Hence, for some p, q, each face is a p-gon, each vertex has valence q, and the map is said to be of type {p, q}.

Coxeter [l] classified all rotary maps on the torus and showed that they fall into three families: (4, 4}b,c, (3, 6}b,o and (6 , 3}b,c, parametrized by b, c so that each map is regular if and only if (iff) bc(b - c ) = 0. Thus there are infinitely many chiral maps on the torus. Sherk [7] and Garbe [4] constructed families of rotary maps by forming branched coverings of the toroidal rotary maps; the coverings being chiral exactly when the base maps are chiral.

Garbe [5] showed that there are no chiral maps of genus 2, 3, 4, 5 , or 6. Edmonds [2, p. 3881 constructed an example of a chiral map of type {7,7} of genus 7 (Fig. 1). This self-dual map has 8 faces, 8 vertices, and Journal of Graph Theory, Vol. 2 (1978) 315-318 @ 1978 by John Wiley & Sons, Inc. 0364-9024/78/0002-0315$01 .OO

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FIGURE 1. Edmond's chiral map of genus 7. This is Heffter's map for q = 8.

FIGURE 2. Chiral branched cover of the regular map (6, 3},,,. This map and its dual are the smallest nontoroidal chiral maps.

NONTOROIDAL CHIRAL MAPS 317

28 edges. Heffter [6] constructed, for each prime power q, a rotary map with underlying graph Kq ; and for q > 4, this map is chiral. For q = 8, it is Edmond’s map. For any n > 6, a rotary embedding of K, must be chiral.

The map M of Figure 2, first discovered by Peter Bergau [5, p. 391, arises as a 3-fold branched covering of the regular map (6, 3}1,1 (Fig. 3). The covering is given by this table:

1 2 3 10 11 12 19 20 21 4 5 6 13 14 15 22 23 24 7 8 9 16 17 18 25 26 27 1 2 3 4 5 6 7 8 9

the three edges of M in one column being projected onto the edge of (6, 3}1,, listed at the bottom of that column.

That M is rotary is shown by displaying the automorphisms R, S as permutations on the edges, where R is rotation one step counterclockwise around face 1 and S is rotation one step clockwise around the central vertex:

R = ( l 2 18 16 23 27) (3 25 14 6 22 17) (4 8 12 13 26 24) (5 15 10 20 21 7) (9 19 11)

S = ( l 2 3 4 5 6 7 8 9) (10 11 12 13 14 15 16 17 18) (19 20 21 22 23 24 25 26 27)

FIGURE 3. Regular map (6, 3},,,.

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In spite of the fact that the base map is regular, the map M is chiral. To see that no reflection of type X is a symmetry of this map, note that such an automorphism would seem to interchange edges 23 and 16 in face 1, and simultaneously interchange 23 and 13 in faces 5 and 6. A chiral 3n-fold covering of (6, 3}1,1 exists for each n.

The map M has 27 edges, just one less than the Heffter-Edmonds map. It is not difficult to show by exhaustion that no map of fewer edges of genus 7 is chiral; in fact, the only chiral maps of any genus with fewer edges are the toroidal maps: (4, 4}2,1, (4, 4}3.1, (4, 4}3,2, (3,6}2,1, and its dual (6, 3}2,1. Thus the map A4 and its dual are the smallest nontoroidal chiral maps (i.e. those with the fewest edges and hence the smallest groups).

ACKNOWLEDGMENT

The author wishes to thank the anonymous referee of an earlier paper for calling Heffter’s paper to his attention.

Note added in proof: N. L. Biggs [Automorphisms of imbedded graphs. J. Combinatorial Theory 11 (1971) 132-1381 has shown that K,, is the underlying graph of a rotary map iff n is a prime power.

References

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[2] H. S . M. Coxeter, Inrroducrion to Geometry. Wiley, New York (1969). [3] H. S. M. Coxeter and W. 0. J. Moser, Generators and Relations for

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[5] D. Garbe, Uber die regularen zerlegungen geschlossener orientier-

[6] L. Heffter, Ueber metacyklische gruppen und nachbar-

[7] F. A. Sherk, A family of regular maps of type {6,6}. Canad. Math.

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