(N,p)-equitable b-coloring of graphs

We present three variants which correspond to the possible cases that we can encounter when solving the problem and we examine and analyze them separately. Furthermore, we prove that the problem of determining the (N,p)-equitable b-coloring of a graph G with N colors is NP-complete. Moreover, we obtain some lower bounds and exact values for trees and we distinguish some particular cases when a pivoted tree or an arbitrary not pivoted tree can be (N,p)-equitable b-colorable with p=0 or p=1. Finally, we summarize, draw conclusions and point out some directions for future work.