Total dominator chromatic number of $k$-subdivision of graphs

TL;DR

This paper investigates the total dominator chromatic number of the k-subdivision of a graph, providing insights into how subdividing edges affects the minimum number of colors needed for such colorings.

Contribution

It introduces the concept of total dominator chromatic number for k-subdivisions and analyzes its properties, offering new theoretical results in graph coloring.

Findings

Determined bounds for the total dominator chromatic number of k-subdivisions.

Established relationships between the original graph and its k-subdivision.

Provided exact values or bounds for specific classes of graphs.

Abstract

Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $\chi_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the total dominator chromatic number of $k$-subdivision of $G$.