Color-character of uncolorable cubic graphs

Let G=(V,E)G=(V,E) be a cubic graph with chromatic index 4 and c:E⟶{0,1,2,3}c:E⟶{0,1,2,3} a proper 4-edge-coloring of GG. Let Ei={e∈E∣c(e)=i}Ei={e∈E∣c(e)=i} and ∘(c)=min{|Ei|∣i=0,1,2,3}∘(c)=min{|Ei|∣i=0,1,2,3}. If C(G)C(G) denotes all the proper 4-edge-colorings of GG, then m(G)=minc∈C(G){∘(c)}m(G)=minc∈C(G){∘(c)} is defined to be the color-character   of GG. In this work, we prove that m(G)m(G) is a constant under some operations, and give a relation between m(G)m(G) and another parameter of GG.