[Paper Review] Equitable chromatic threshold of Kronecker products of complete graphs

Zhidan Yan, Wei Wang|arXiv (Cornell University)|Aug 4, 2012

Advanced Graph Theory Research8 references3 citations

TL;DR

This paper determines the equitable chromatic threshold $\chi_=^*(K_m \times K_n)$ for Kronecker products of complete graphs $K_m$ and $K_n$ with $n \geq m \geq 2$. It establishes a closed-form expression depending on the residue of $n$ modulo $m+1$, distinguishing cases based on whether $n \equiv 0,1 \pmod{m+1}$ or $n \equiv 2,\dots,m \pmod{m+1}$, with the threshold given by $\lceil \frac{mn}{m+1} \rceil$ or $m\lceil \frac{n}{s^*} \rceil$, where $s^*$ is the smallest divisor of $n$ that is at least $m+2$. The result fully characterizes equitable colorability thresholds for these graph products.

ABSTRACT

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic threshold of a graph $G$, denoted by $χ_=^*(G)$, is the minimum $k$ such that $G$ is equitably $k^\prime$-colorable for all $k^\prime \ge k$. Let $G imes H$ denote the direct product of graphs $G$ and $H$. For $n\ge m\ge 2$ we prove that $χ_=^*(K_{m} imes K_n)$ equals $\lceil\frac{mn}{m+1} ceil$ if $n\equiv 2,...,m ( extup{mod} m+1)$, and equals $m\lceil\frac{n}{s^\star} ceil$ if $n\equiv 0,1 ( extup{mod} m+1)$, where $s^\star$ is the minimum positive integer such that $s^\star mid n$ and $s^\star\ge m+2.$

Motivation & Objective

• To determine the equitable chromatic threshold $\chi_=^*(K_m \times K_n)$ for Kronecker products of complete graphs with $n \geq m \geq 2$.
• To characterize the minimum $k$ such that $K_m \times K_n$ is equitably $k'$-colorable for all $k' \geq k$.
• To resolve the threshold behavior under different modular conditions of $n$ modulo $m+1$, particularly distinguishing cases where $n \equiv 0,1 \pmod{m+1}$ versus $n \equiv 2,\dots,m \pmod{m+1}$.
• To introduce and utilize the parameter $s^*$, the smallest divisor of $n$ that is at least $m+2$, in the threshold formula for specific residue classes.

Proposed method

• The authors analyze the structure of the Kronecker product $K_m \times K_n$, which has $mn$ vertices and is $ (m-1)(n-1) $-regular.
• They apply combinatorial arguments based on equitable coloring constraints, ensuring color class sizes differ by at most one.
• The threshold is derived by examining the minimal $k$ such that $mn$ vertices can be partitioned into $k$ color classes of size at most $\lceil \frac{mn}{k} \rceil$, with the maximum class size minimized.
• For $n \equiv 2,\dots,m \pmod{m+1}$, the threshold is shown to be $\lceil \frac{mn}{m+1} \rceil$ via bounding the size of the largest color class.
• For $n \equiv 0,1 \pmod{m+1}$, the threshold is determined using the minimal divisor $s^*$ of $n$ with $s^* \geq m+2$, leading to $m \lceil \frac{n}{s^*} \rceil$.
• The proof involves case analysis based on modular arithmetic and the properties of divisors to ensure equitable colorability is achievable from the threshold onward.

Experimental results

Research questions

• RQ1 What is the equitable chromatic threshold $\chi_=^*(K_m \times K_n)$ for $K_m \times K_n$ when $n \geq m \geq 2$?
• RQ2 How does the threshold depend on the residue of $n$ modulo $m+1$, particularly distinguishing $n \equiv 0,1 \pmod{m+1}$ from $n \equiv 2,\dots,m \pmod{m+1}$?
• RQ3 Why is the parameter $s^*$, the smallest divisor of $n$ with $s^* \geq m+2$, critical in determining the threshold for $n \equiv 0,1 \pmod{m+1}$?
• RQ4 Can the equitable chromatic threshold be expressed in closed form for all $n \geq m \geq 2$?
• RQ5 Is the threshold $\lceil \frac{mn}{m+1} \rceil$ sufficient and necessary for equitable $k'$-colorability when $n \equiv 2,\dots,m \pmod{m+1}$?

Key findings

• The equitable chromatic threshold $\chi_=^*(K_m \times K_n)$ is $\lceil \frac{mn}{m+1} \rceil$ when $n \equiv 2,\dots,m \pmod{m+1}$.
• When $n \equiv 0,1 \pmod{m+1}$, the threshold is $m \lceil \frac{n}{s^*} \rceil$, where $s^*$ is the smallest divisor of $n$ such that $s^* \geq m+2.$
• The threshold is always an integer and is the minimal $k$ such that $K_m \times K_n$ is equitably $k'$-colorable for all $k' \geq k$.
• The threshold expression depends critically on the modular class of $n$ modulo $m+1$, indicating a structural phase change in the colorability behavior.
• For $n \equiv 0,1 \pmod{m+1}$, the value $s^*$ ensures that the color class sizes remain balanced under the equitable constraint.
• The results fully characterize the equitable chromatic threshold for all $K_m \times K_n$ with $n \geq m \geq 2$, resolving an open problem in equitable coloring of graph products.