Edge covered critical multigraphs

Let GG be a multigraph with edge set E(G)E(G). An edge coloring CC of GG is called an edge covered 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 covered coloring is called the edge covered chromatic index of GG and is denoted by χc′(G). A graph GG is said to be of class  CI if χc′(G)=δ and otherwise of class  CII. A pair of vertices {u,v}{u,v} is said to be critical   if χc′(G+uv)>χc′(G). A graph GG is said to be edge covered critical   if it is of class CII and every edge with vertices in V(G)V(G) not belonging to E(G)E(G) is critical. Some properties about edge covered critical graphs are considered.