An Analogy Between Edge Colourings and Differentiable Manifolds, with a New Perspective on 3-Critical Graphs

Abstract

A graph or more generally a multigraph can be interpreted as a family of stars—one star for each vertex—which adequately intersect on certain edges, so as to generate a global adjacency structure. An edge colouring can be read as an injective assignment of colours to each star, enjoying a “compatibility” property on adjacent vertices: for, any two intersecting stars must obviously get the same colour on each pair of overlapping edges (stars of multigraphs may have more than one overlap). The above interpretation justifies some key definitions which make an edge colouring rather similar to a differentiable atlas on a manifold. In the case of simple graphs, the distinction between class 1 and class 2 becomes the distinction between orientable and non-orientable atlases. In particular, $k$-critical graphs with $2\le k\le 3$ are shown to be, in most cases, the result of an identification of extremal edges or vertices which is analogous to the topological identification yielding the Möbius strip from the rectangular strip. Moving along the strip is equivalent to transmitting a fixed colour across the local charts (stars) of the graph. Accordingly, we revisit the known classification of small 3-critical graphs, with a specific stress on the various types of graphs which lose orientability (i.e. become critical) after the identification of their extremes.