Bilangan Kromatik Graceful pada Keluarga Graf Unicyclic

Abstract

Suppose $ G $ is a graph, where $ G = \{V (G), E (G) \} $. Graceful coloring is defined by $ c: V (G) \to \{1,2, ..., k \} $ which induces a proper edge coloring $ c': E (G) \to \{1,2, ..., k- 1 \}$ defined by $c'(xy)=|c(x)-c(y)|$, where $ k \geq 2 $, $ k \in N $. Coloring is said to be graceful if these 3 conditions are satisfied, namely the proper vertex color, the proper edge color, and the edge color, which are the absolute difference between the color of the accident vertex. The subgraph $H$ on that graceful coloring is smaller than the $G$. Furthermore, one of the subgraphs in the unicyclic graph family is a cycle graph. The graceful chromatic number on a graph denoted by $ \chi_g (G) $, is the optimum number of graceful colors from graph $G$. This research aims to find graceful chromatic numbers in the unicyclic graph family, namely bull graphs, net graphs, cricket graphs, caveman graphs, peach graphs, and flowerpot graphs. The results of this study indicate that $\chi_g(C_l) \geq 4$, where $C_l$ is a unicyclic graphs.