An odd independent set S in a graph G= (V, E) is an independent set of vertices such that, for every vertex v V S, either N (v) S = or |N (v) S| 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₎₃ (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 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 χₒ₎ (G). A simple relation involving these two parameters and the order |G| of G is α₎₃ (G) χₒ₎ (G) |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 ₎₃ (P_ P_) 0. 384615. . . , where ₎₃ (P_ P_) is the odd independence ratio. (2) χₒ₎ (Gd) = 3 for all d 1, where Gd is the infinite d-dimensional grid. As a consequence, ₎₃ (Gd) 1/3. (3) The r-King graph G on n² vertices has α₎₃ (G) = n/ (2r+1) ². Moreover, χₒ₎ (G) = (2r + 1) ² if n 2r + 1, and χₒ₎ (G) = n² if n 2r. Many open problems are given for future research.
Caro et al. (Thu,) studied this question.