Color-character of uncolorable cubic graphs

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