The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs

Abstract

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $\alpha_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $\chi_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $\alpha_{od}(G)\cdot\chi_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. In the present work, which is a companion to our first paper on the subject [The odd independence number of graphs, I: Foundations and classical classes], we focus on grid-like and chessboard-like graphs and compute or estimate their odd independence number and their strong odd chromatic number. Among the many results obtained, the following give the flavour of this paper: (1) $0.375 \leq \varrho_{od}(P_\infty \Box P_\infty) \leq 0.384615...$, where $\varrho_{od}(P_\infty \Box P_\infty)$ is the odd independence ratio. (2) $\chi_{so}(G_d) = 3$ for all $d \geq 1$, where $G_d$ is the infinite $d$-dimensional grid. As a consequence, $\varrho_{od}(G_d) \geq 1/3$. (3) The $r$-King graph $G$ on $n^2$ vertices has $\alpha_{od}(G) = \lceil n/(2r+1) \rceil^2$. Moreover, $\chi_{so}(G) = (2r + 1)^2$ if $n \geq 2r + 1$, and $\chi_{so}(G) = n^2$ if $n \leq 2r$. Many open problems are given for future research.