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

Abstract

Il problema della colorabilità di spigoli di grafi critici viene riletto col linguaggio della geometria differenziale; in particolare la distinzione tra classe 1 e 2 diventa la nota distinzione tra presenza e assenza di orientabilità

A Möbius-type gluing technique for obtaining edge-critical graphs

“…Following the mentioned approach in [11], we also show that the infinite family of Goldberg's graphs disproving the "Critical Graph Conjecture" and the other two counterxamples constructed by Fiol and Chetwynd can be obtained by a suitable identification of vertices which is pretty analogous to the topological identification yielding the Möbius strip from a rectangular strip. Details about the change of language -from topology to graph theory -can be found in [11].…”

“…Similarly, adding one pendant edge uw when u and v are saturating yields conflicting vertices w, v. Therefore, the above operations can be regarded as identifications of conflicting vertices in all cases. These procedures could be rephrased in terms of atlases and orientability, as explained in [11]; the prototype of this analogy is given by the odd cycle C 2n+1 of any fixed length. Such a graph is the result of the identification of two conflicting vertices, namely, the extremes of the path P 2n+2 having the same number of edges.…”

“…In this paper, we push forward the analogy between non-orientable manifolds and class 2 graphs which was introduced in [11] and describe a new method for constructing critical graphs. We show the effectiveness of this method by constructing infinite families of critical simple graphs.…”

An improvement to the vertex-splitting conjecture

“…We say G is obtained from G by a vertex-splitting. A vertexsplitting was formulated in terms of the "Mőbius-type gluing technique" in [1] and [7]. Hilton and Zhao [3] in 1997 propsed the following conjecture.…”