A method to optimize antipodal coloring span of graphs and its application

In this article, we study radio $k$-colorings of simple connected graphs $G$ with diameter $d$, where a radio $k$-coloring $g$ assigns non-negative integers to $V(G)$ (vertices of $G$) such that $|g(u) - g(v)| \geq 1 + k - d(u, v)$ for any two vertices $u, v$ with $1 \leq k \leq d$. The span of a radio $k$-coloring $g$, expressed by $rc_k(g)$, is the maximum integer assigned by $g$, and the radio $k$-chromatic number $rc_k(G)$ is the minimum span among all radio $k$-colorings of $G$. A coloring $g$ is minimal if $rc_k(g) = rc_k(G)$. When $k = d-1$, this coloring is known as the antipodal coloring, and $rc_{d-1}(G)$ referred to as the antipodal number, is denoted by $ac(G)$. We derive a sufficient condition for an antipodal coloring to be minimal and apply this criterion to determine the antipodal number of the generalized Petersen graph $GP(n,1)$ for all $n$ except when $n \equiv 2 \pmod{8}$, and for toroidal grids $T_{r,s} = C_r \square C_s$ when $rs$ is even. Additionally, we establish a lower bound for $ac(T_{r,s})$ when $rs$ is odd.