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

TL;DR

The paper investigates the odd independence number $α_{od}(G)$ and the strong odd chromatic number $χ_{so}(G)$, focusing on grid- and chessboard-like graphs, and establishes both general bounds and exact values in many cases. It proves $α_{od}$ is supermultiplicative under Cartesian product and provides density bounds for infinite grids, including $ρ_{od}(P_\infty\Box P_\infty)$ lying between $3/8$ and $5/13$, and shows $χ_{so}=3$ for all infinite $d$-dimensional grids, with $α_{od}$ densities bounded away from zero. In planar and higher-dimensional grids, it determines or tightly bounds $α_{od}$ and $χ_{so}$ for several finite and infinite configurations (e.g., King, Rook, Bishop, Knight, and Queen graphs), including explicit strong odd colorings (e.g., a 9-color coloring for kings and a 3-color pattern for infinite grids). The work also surveys a broad set of open problems and conjectures, linking odd independence to classical domination and coloring parameters and highlighting several intriguing questions about densities and colorings on diverse grid tilings 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 $α_{od}(G)$, 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 $χ_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $α_{od}(G)\cdotχ_{so}(G) \geq |G|$, 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) $χ_{so}(G_d) = 3$ 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 $α_{od}(G) = \lceil n/(2r+1) \rceil^2$. Moreover, $χ_{so}(G) = (2r + 1)^2$ if $n \geq 2r + 1$, and $χ_{so}(G) = n^2$ if $n \leq 2r$. Many open problems are given for future research.

Figures (12)

• Figure 1: $\alpha_\mathrm{od}(C_4\,\Box\, C_4)=6$.
• Figure 2: Detail of independent set of density $3/8$ in $P_\infty\,\Box\, P_\infty$, generated from $\alpha_\mathrm{od}(C_4\,\Box\, C_4)=6$.
• Figure 3: Cellular representation, also indicating the generating pattern which yields $\varrho_\mathrm{od}\geq 3/8$.
• Figure 4: Strong odd 3-coloring of $P_\infty\,\Box\, P_\infty$, cellular representation. (a) Odd coloring in the first coordinate, (b) modified coloring in levels of odd index.
• Figure 5: A $7\times 7$ square table (left), and its frame (right).

Theorems & Definitions (48)

• Proposition 1
• proof
• Example 2
• Proposition 3
• proof
• Definition 4
• Lemma 5: part-1
• Proposition 6
• proof
• Proposition 7