On metric chromatic number of comb product of ladder graph

All graphs in this paper are connected and nontrivial graph. Let f : V(G) → {1,2,…,k} be a vertex coloring of a graph G where two adjacent vertices may be colored the same color. Consider the color classes Π = {C 1 , C 2 ,…, C k }. For a vertex v of G, the representation color of v is the k-vektor r(v | Π) = {d(v, C 1 ),d(v, C 2 ),…, d(v, C k )}, where d(v, Ci) = min{ d(v, c); c e C i }. If r(u | Π) = r(v | Π) for every two adjacent vertices u and v of G, then f is a metric coloring of G . The minimum k for which G has a metric k -coloring is called the metric chromatic number of G and is denoted by μ(G). The metric chromatic number on comb product of ladder graphs namely path graph, star graph, fan graph, cycle graph, and complete graph.