On bounds for the incidentor chromatic number of a directed weighted multigraph

An incidentor coloring of a directed weighted multigraph is called admissible if: (a) the incidentors adjoining the same vertex are colored by different colors; (b) the difference between the colors of the final and initial incidentors of each arc is at least the weight of this arc. The minimum number of colors necessary for an admissible coloring of all incidentors of a multigraph $G$ is bounded above and below. The upper and lower bounds differ by $\lceil\Delta/2\rceil$ where $\Delta$ is the degree of $G$.