Two coloring problems on matrix graphs

TL;DR

This paper introduces matrix graphs based on finite fields, explores their properties, and solves specific coloring problems relevant to optical network scalability.

Contribution

It defines a new class of matrix graphs, determines the exact chromatic number for certain coloring constraints, and provides bounds for others.

Findings

Exact value of '_d(N n, q) determined.

Upper and lower bounds for _d(N n, q) established.

New insights into coloring problems on matrix graphs.

Abstract

In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two matrices share an edge if the rank of their difference is $1$. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let $\chi'_d(N\times n, q)$ (resp. $\chi_d(N\times n, q)$) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most $d$ (resp. exactly $d$) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of $\chi'_d(N\times n,q)$ and give some upper and lower bounds on $\chi_d(N\times n,q)$.