on Graceful Chromatic Number of Vertex amalgamation of Tree Graph Family

Abstract

Definition graceful k-coloring of graph 𝐺 = (𝑉, 𝐸) is proper vertex coloring 𝑐: 𝑉(𝐺) → {1,2, … , 𝑘); 𝑘 ≥ 2, which induces a proper edge coloring 𝑐′: 𝐸(𝐺) → {1,2, … , 𝑘 − 1} defined 𝑐 ′ (𝑢𝑣) = |𝑐(𝑢) − 𝑐(𝑣)|. The minimum vertex coloring from graph 𝐺 can be colored with graceful coloring called a graceful chromatic number with notation 𝜒𝑔 (𝐺). The method used in this research is the axiomatic deductive method and pattern recognition. The detection method finds coloring patterns and graceful chromatic numbers based on the amalgamation operation of point family tree graphs. In this paper, we have investigated the graceful chromatic number of vertex amalgamation of tree graph family: path graph, centipede graph, broom, and E graph. The results of this study are expected to be used as a basis for studies in developing science and applications related to graceful chromatic numbers on the results of point amalgamation operations in tree family graphs.