Vertex and edge coloring have their diverse applications in problems such as time tabling and scheduling, frequency assignment for spectrum, register allocation, pattern matching, analysis of biological and archeological data, etc. An l-vertex-coloring is a generalized version of the vertex coloring of a graph with integers that asks assigning colors to vertices such that any two vertices u and v get different colors if dist(u,v) is at most l, where dist(u,v) denotes the length of the shortest path between u and v in G, l being a nonnegative integer. A coloring is optimal if it uses minimum...