Superposition of snarks revisited

Abstract

In this paper we propose a new approach to superposition of snarks, a powerful method of constructing large cubic graphs with no 3-edge-colouring from small ones. The main idea is to use surjective mappings between graphs similar to graph homomorphisms and to control flows induced from the domain graph to the target graph via the mappings. This leads to significant strengthening of the power of the classical superposition, which we illustrate by several examples and two applications. First, we describe a family of cyclically 5-edge-connected snarks Gd of order 3⋅2d−2, with d≥2, each being spanned by the balanced cubic tree of depth d; the family contains the Petersen graph as G2. The existence of such a family was conjectured by Hoffmann-Ostenhof and Jatschka (2017). Second, we construct cyclically 5-edge-connected permutation snarks of order n for each n≡2(mod8) with n≥34. Our construction employs a rather exotic form of superposition where the proof of uncolourability of the resulting graph requires two graphs derived from the target graph to be snarks rather than just the target graph itself.