Analisa Pewarnaan Total r-Dinamis pada Graf Lintasan dan Graf Hasil Operasi

Abstract

Graph coloring began to be developed into coloring dynamic. One of the developments of dynamic coloring is $r$-dynamic total coloring. Suppose $G=(V(G),E(G))$ is a non-trivial connected graph. Total coloring is defined as $c:(V(G) \cup E(G))\rightarrow {1,2,...,k}, k \in N$, with condition two adjacent vertices and the edge that is adjacent to the vertex must have a different color. $r$-dynamic total coloring defined as the mapping of the function $c$ from the set of vertices and edges $(V(G)\cup E(G))$ such that for every vertex $v \in V(G)$ satisfy $|c(N(v))| = min{[r,d(v)+|N(v)|]}$, and for each edge $e=uv \in E(G)$ satisfy $|c(N(e))| = min{[r,d(u)+d(v)]}$. The minimal $k$ of color is called $r$-dynamic total chromatic number denoted by $\chi^{\prime\prime}(G)$. The $1$-dynamic total chromatic number is denoted by $\chi^{\prime\prime}(G)$, chromatic number $2$-dynamic denoted with $\chi^{\prime\prime}_d(G)$ and $r$-dynamic chromatic number denoted by $\chi^{\prime\prime}_r(G)$. The graph that used in this research are path graph, $shackle$ of book graph $(shack(B_2,v,n)$ and \emph{generalized shackle} of graph \emph{friendship} $gshack({\bf F}_4,e,n)$.