The classification of combinatorial surfaces using 3-graphs

A 3-graph is a cubic graph endowed with a proper edge colouring in three colours. Surfaces can be modelled by means of 3-graphs. We show how 3-graphs can be used to establish the standard classification of sUrfaces by orientability and Euler characteristic. In [8], Tutte approaches topological graph theory from a combinatorial viewpoint. In particular, an entirely combinatorial approach to the classification of sUli'aces is given. He uses the idea of a premap, which is expressed in [6] as a special kind of 3-graph. In the next section, we define 3-graphs as cubic graphs endowed with a proper edge colouring in three colours. They are also studied in [3, 9] in the more general setting of n-graphs, a variation of the traditional simplicial complex approach to algebraic topology. In fact, the classification of surfaces in terms of 3-graphs is a direct consequence of the main theorem in [3], and is explicitly stated in [10]. Our purpose here is to show how the classification of surfaces by means of 3-graphs follows from Tutte's approach in [8] and the relationship between 3-graphs and premaps. We shall find that this approach provides a possible tool for proving theorems about cubic graphs with a proper edge colouring in three colours. Since most of this paper consists of reproving a known result, it may be classed as expository. Throughout this paper, the sum of sets is defined as their symmetric difference.