Pewarnaan Titik r-Dinamis pada Keluarga Graf Unicyclic sebagai E-Monograf

Abstract

A graph is defined as an ordered set (𝑉, 𝐸) where 𝑉 is a non-empty set of elements called vertices and 𝐸 is a set of edges which are finite and may be empty and each edge connects two different points of 𝑉(𝐺). The 𝑟-dynamic coloring is defined as 𝑐: 𝑉(𝐺) → {1,2,3, … , 𝑘}such that it satisfies the following conditions if 𝑢𝑣 ∈ 𝐸(𝐺), then 𝑐(𝑢) ≠ 𝑐(𝑣), and ∀𝑣 ∈ 𝑉(𝐺), |𝑐(𝑁(𝑣))| ≥ 𝑚𝑖𝑛{𝑟, 𝑑(𝑣)}. The notation 𝑁(𝑣) on a graph can be interpreted as a point around 𝑣, whereas |𝑐(𝑁(𝑣))| is the number of coloring functions around point 𝑣. The 𝑟-dynamic coloring must satisfy |𝑐(𝑁(𝑣))| ≥ 𝑚𝑖𝑛{𝑟, 𝑑(𝑣)}, where 𝑚𝑖𝑛{𝑟, 𝑑(𝑣)} is a minimum between 𝑟 and the degree of point 𝑣. The function that maps a point 𝑢 on a graph to the color set {1,2,3, . . , 𝑘} in 𝑟-dynamic coloring can be donated as 𝑐(𝑢). The vertices that are connected to each other in the graph must have a different color. The purpose of 𝑟-dynamic coloring is to find the minimum chromatic number of graph coloring with unlimited parameter 𝑟. Dynamic coloring is performed on unicyclic graph families or grouped graphs because they have the same charcteristics, namely consisting of one circle graph. Unicyclic graph families, especially cricket graphs, peach graphs, and flowerpot graphs were chosen because no previous research had been carried out. The proof begins with determining the lower limit, then determining the upper limit with the coloring function, and checking the 𝑟- dynamic coloring function obtained with the help of the table so that it conforms to the definition of 𝑟-dynamic coloring.