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

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.