Let f be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ(v) = (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) : x ∈ Vi}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then f...

Let c be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ(v) := (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) | x ∈ Vi}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then ...

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

Let c be a proper k-coloring of a connected graph G andΠ = (C1, C2, . . . , Ck) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c Π (v) := (d(v, C1), d(v, C2), . . . , d(v, Ck)), where d(v, Ci) = min{d(v, x)|x ∈ Ci}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c...

The locating-chromatic number of a graph can be defined as the cardinality of a minimum resolving partition of the vertex set such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in are not contained in the same partition class. In this case, the coordinate of a vertex in is expressed in terms of the distances of to all partition classe...

Let G be a simple graph and (G,) denotes the number of proper vertex colourings of G with at most  colours, which is for a fixed graph G , a polynomial in  , which is called the chromatic polynomial of G . Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.