On graphs whose domination number is equal to chromatic and dominator chromatic numbers

TL;DR

This paper investigates graphs where the domination number, chromatic number, and dominator chromatic number are all equal, providing existence results for such graphs of certain sizes and characterizing planar cases.

Contribution

It establishes the existence of $D(k)$ graphs for large enough orders and characterizes planar $D(k)$ graphs as bipartite complete graphs $K_{2,q}$.

Findings

Existence of $D(k)$ graphs for all $n geq 4k - 3$

No planar $D(k)$ graphs for $k otin ext{{2}}$

Planar $D(k)$ graphs are exactly $K_{2,q}$ with $q \\geq 2$

Abstract

For a graph $G = (V(G), E(G))$, a dominating set $D$ is a vertex subset of $V(G)$ in which every vertex of $V(G) \setminus D$ is adjacent to a vertex in $D$. The domination number of $G$ is the minimum cardinality of a dominating set of $G$ and is denoted by $\gamma(G)$. A coloring of $G$ is a partition $C = (V_{1}, ... ,V_{k})$ such that each of $V_{i}$ in an independent set. The chromatic number is the smallest $k$ among all colorings $C = (V_{1}, ... ,V_{k})$ of $G$ and is denoted by $\chi(G)$. A coloring $C = (V_{1}, ... ,V_{k})$ is said to be dominator if, for all $V_{i}$, every vertex $v \in V_{i}$ is singleton in $V_{i}$ or is adjacent to every vertex of $V_{j}$. The dominator chromatic number of $G$ is the minimum $k$ of all dominator colorings of $G$ and is denoted by $\chi_{d}(G)$. Further, a graph $G$ is $D(k)$ if $\gamma(G) = \chi(G) = \chi_{d}(G) = k$. In this paper, for $n \geq 4k - 3$, we prove that there always exists a $D(k)$ graph of order $n$. We further prove that there is no planar $D(k)$ graph when $k \in \{3, 4\}$. Namely, we prove that, for a non-trivial planar graph $G$, the graph $G$ is $D(k)$ if and only if $G$ is $K_{2, q}$ where $q \geq 2$.