Two coloring problems on matrix graphs

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 $\chi_d(N\times n,q)$.Comment: 9 page

New Results on Two Hypercube Coloring Problems

In this paper, we study the following two hypercube coloring problems: Given $n$ and $d$, find the minimum number of colors, denoted as ${\chi}'_{d}(n)$ (resp. ${\chi}_{d}(n)$), needed to color the vertices of the $n$-cube such that any two vertices with Hamming distance at most $d$ (resp. exactly $d$) have different colors. These problems originally arose in the study of the scalability of optical networks. Using methods in coding theory, we show that ${\chi}'_{4}(2^{r+1}-1)=2^{2r+1}$, ${\chi}'_{5}(2^{r+1})=4^{r+1}$ for any odd number $r\geq3$, and give two upper bounds on ${\chi}_{d}(n)$. The first upper bound improves on that of Kim, Du and Pardalos. The second upper bound improves on the first one for small $n$. Furthermore, we derive an inequality on ${\chi}_{d}(n)$ and ${\chi}'_{d}(n)$.Comment: The material in this paper was presented at The Fifth Shanghai Conference on Combinatorics, May 14-18, 2005, Shanghai, China. This paper has been submitted for publicatio

New bounds on a hypercube coloring problem and linear codes

In studying the scalability of optical networks, one problem arising involves coloring the vertices of the n-dimensional hypercube with as few colors as possible such that any two vertices whose Hamming distance is at most k are colored differently. Determining the exact value of O/_k(n), the minimum number of colors needed, appears tobe a difficult problem. In this paper, we improve the known An n-cube (or n-dimensional hypercube) is a graph whose vertices are the vectors of th