On The Number of Edge-3-Colourings of A Snipped Snark

2013/04/19 by McKinney, Scott A.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.5427

Abstract

For a given snark G and edge e of G, we can form a cubic graph G_e using an operation we call "edge subtraction". The number of 3-edge-colourings of G_e is 18 * ψ(G,e) for some nonnegative integer ψ(G,e). Given snarks G₁ and G₂, we can form a new snark G using techniques given by Isaacs and Kochol. In this note we give relationships between ψ(G₁,e₁), ψ(G₂,e₂), and ψ(G,e) for particular edges e₁, e₂, and e, in G₁, G₂, and G (respectively). As a consequence, if g,h,i,j,k,l are each a nonnegative integer, then there exists a cyclically 5-edge-connected snark G with an edge e such that ψ(G,e)=5^g * 7^h, and a cyclically 4-edge-connected snark G₀ with an edge e₀ such that ψ(G₀,e₀)=2ⁱ * 3^j * 5^k * 7^l.

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