A note on reducing resistance in snarks

Abstract

In this paper, we define new parameters which measure the edge uncolourability of cubic graphs. These parameters count the minimum number of vertex reductions required to render a 3-edge-colourable graph, given a cubic graph G. Two of these parameters we denote as  1_r(G) and e_r(G). We recall as well that the resistance of G, denoted as r(G), counts the minimum number of edges which can be  removed from G in order to render 3-edge-colourability. Also, the weak oddness of G denoted as ω¹(G) counts the minimum number of  odd components in an even factor of G, and the flow resistance of G denoted as r_f (G) counts the minimum number of zero edges in a  4-flow of G. Weak oddness and flow resistance are also considered measurements of edge uncolourability in cubic graphs. We conjecture  that r(G) ≥ 1_r(G) and that r(G) ≥ e_r(G). We then prove some significant consequences of these conjectures being true with regards to  weak oddness and flow resistance.