First-Fit coloring of Cartesian product graphs and its defining sets

Let the vertices of a Cartesian product graph $G\Box H$ be ordered by an ordering $\sigma$. By the First-Fit coloring of $(G\Box H, \sigma)$ we mean the vertex coloring procedure which scans the vertices according to the ordering $\sigma$ and for each vertex assigns the smallest available color. Let $FF(G\Box H,\sigma)$ be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether $FF(G\Box H,\sigma)=FF(G\Box H,\tau)$, where $\sigma$ and $\tau$ are arbitrary orders. We study and obtain some bounds for $FF(G\Box H,\sigma)$, where $\sigma$ is any quasi-lexicographic ordering. The First-Fit coloring of $(G\Box H, \sigma)$ does not always yield an optimum coloring. A greedy defining set of $(G\Box H, \sigma)$ is a subset $S$ of vertices in the graph together with a suitable pre-coloring of $S$ such that by fixing the colors of $S$ the First-Fit coloring of $(G\Box H, \sigma)$ yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of $G\Box H$ with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in $G\Box H$, including some extremal results for Latin squares.Comment: Accepted for publication in Contributions to Discrete Mathematic

Critical sets in the elementary abelian 2- and 3- groups

In 1998, Khodkar showed that the minimal critical set in the Latin square corresponding to the elementary abelian 2-group of order 16 is of size at most 124. Since the paper was published, improved methods for solving integer programming problems have been developed. Here we give an example of a critical set of size 121 in this Latin square, found through such methods. We also give a new upper bound on the size of critical sets of minimal size for the elementary abelian 2-group of order $2^n$: $4^{n}-3^{n}+4-2^{n}-2^{n-2}$. We speculate about possible lower bounds for this value, given some other results for the elementary abelian 2-groups of orders 32 and 64. An example of a critical set of size 29 in the Latin square corresponding to the elementary abelian 3-group of order 9 is given, and it is shown that any such critical set must be of size at least 24, improving the bound of 21 given by Donovan, Cooper, Nott and Seberry.Comment: 9 page

Intercalates and Discrepancy in Random Latin Squares

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely $N\ge\left(1-o\left(1\right)\right)\,n^{2}/4$, and that $\mathbb{E}N\le\left(1+o\left(1\right)\right)\,n^{2}/2$ (therefore asymptotically almost surely $N\le fn^{2}$ for any $f\to\infty$). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low-discrepancy Latin squares