Bounds for the incidentor chromatic number of a weighted undirected multigraph

A proper incidentor coloring of an undirected weighted multigraph is called admissible if the absolute value of the difference between the colors of the incidentors of each edge is at least the weight of this edge. The minimum number of colors necessary for an admissible incidentor coloring is called the incidentor chromatic number of the multigraph. The problem of finding this number is studied in the paper. The NP-hardness of this problem is proved for Δ colors. Some upper and lower bounds are found for the incidentor chromatic number.