The number of distinguishing colorings of a Cartesian product graph

Abstract

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold $θ(G)$ of a graph $G$ is the minimum number of colors $k$ required that any arbitrary $k$-coloring of $G$ is distinguishing. In this paper, we calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.

Figures (4)

• Figure 1: The graph $G_1= P_4 \Box P_5$ of Example \ref{['IdNotEnough']}.
• Figure 2: The graph $G_2 = P_5 \Box P_6$ of Example \ref{['AllRedundant']}.
• Figure 3: The graph $G_3 = P_4 \Box P_5$ of Example \ref{['dist-ex']}.
• Figure 4: (Left) A coloring of $P_4 \Box P_4$ that is preserved by the two reflections over $P_4$. Any such coloring is also preserved by the rotation of $180^\circ$. (Middle) A coloring of $P_4 \Box P_4$ that is preserved by the two reflections over corner vertices. Any such coloring is also preserved by the rotation of $180^\circ$. (Right) A coloring of $P_4 \Box P_4$ that is preserved by the reflection over vertical $P_4$ and the reflection over down-left and up-right corner vertices. Any such coloring is also preserved by the rotation of $90^\circ$.

Theorems & Definitions (8)

• proof
• Example 3.1
• Example 3.2
• Example 3.3
• proof
• proof
• proof
• proof