Maps on surfaces with boundary.

*(English)*Zbl 0555.05033A map is defined topologically as a 2-cell imbedding of a graph on a (2- dimensional) surface. It has long been known that a map on a closed orientable surface is essentially determined by properties which are strictly combinatorial [J. Edmonds, A combinatorial representation for polyhedral surfaces, Notices Am. Math. Soc. 7, 646 (1960)] and formally proved in [G. A. Jones and the author, Proc. Lond. Math. Soc., III. Ser. 37, 273-307 (1978; Zbl 0391.05024)]. A combinatorial representation of a map on a closed surface which is not necessarily orientable appears in [W. T. Tutte, What is a map?, New Direct. Theory Graphs, Proc. third Ann. Arbor Conf., Univ. Michigan 1971, 309-325 (1973; Zbl 0258.05105)]. This model is extended to surfaces which are not necessarily closed in [R. P. Bryant and the author, Foundations of the theory of maps on surfaces with boundary, Q. J. Math. (to appear)], where the number of boundary components, the orientability and the genus of the imbedding surface of a map are determined from the combinatorial model of the map.

The article under review notes that the determination of the number of boundary components in [Bryant and the author, loc. cit.] ’used a result which is both complicated and at the present time rather inaccessible’ and offers a direct proof of the necessary theorem. The other results in [Bryant and the author, loc. cit.] are summarized, so that the present article gives a simple and self-contained description of a combinatorial model of a map on a general surface.

The article under review notes that the determination of the number of boundary components in [Bryant and the author, loc. cit.] ’used a result which is both complicated and at the present time rather inaccessible’ and offers a direct proof of the necessary theorem. The other results in [Bryant and the author, loc. cit.] are summarized, so that the present article gives a simple and self-contained description of a combinatorial model of a map on a general surface.

Reviewer: T.R.S.Walsh

##### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

57M15 | Relations of low-dimensional topology with graph theory |