A note on the edge cover chromatic index of multigraphs

Let GG be a multigraph with vertex set V(G)V(G). An edge coloring CC of GG is called an edge-cover-coloring if each color appears at least once at each vertex v∈V(G)v∈V(G). The maximum positive integer kk such that GG has a kk-edge-cover-coloring is called the edge cover chromatic index of GG and is denoted by χc′(G). It is well known that min{d(v)−μ(v):v∈V}≤χc′(G)≤δ(G), where μ(v)μ(v) is the multiplicity of vv and δ(G)δ(G) is the minimum degree of GG. We improve this lower bound to δ(G)−1δ(G)−1 when 2≤δ(G)≤52≤δ(G)≤5. Furthermore we show that this lower bound is best possible.