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

TL;DR

This paper investigates the number of 3-edge-colourings in snarks after an operation called edge subtraction, establishing relationships between these counts in different snarks and demonstrating the existence of snarks with specific colouring properties.

Contribution

It provides new formulas relating colouring counts of snarks formed via known constructions and shows the existence of snarks with prescribed colouring multiplicative structures.

Findings

Relationships between -edge-colouring counts in different snarks

Existence of snarks with -edge-colouring counts as products of prime powers

Construction methods for snarks with specific colouring properties

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 * \psi(G,e) for some nonnegative integer \psi(G,e). Given snarks G_1 and G_2, we can form a new snark G using techniques given by Isaacs and Kochol. In this note we give relationships between \psi(G_1,e_1), \psi(G_2,e_2), and \psi(G,e) for particular edges e_1, e_2, and e, in G_1, G_2, 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 \psi(G,e)=5^g * 7^h, and a cyclically 4-edge-connected snark G_0 with an edge e_0 such that \psi(G_0,e_0)=2^i * 3^j * 5^k * 7^l.