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

TL;DR

This paper investigates radio k-colorings of graphs, focusing on antipodal colorings, providing conditions for minimality, and determining the antipodal number for specific graph classes like generalized Petersen graphs and toroidal grids.

Contribution

It introduces a sufficient condition for minimal antipodal colorings and applies it to compute the antipodal number for certain classes of graphs, extending previous knowledge.

Findings

Derived a sufficient condition for minimal antipodal colorings.

Determined the antipodal number for generalized Petersen graphs except when n ≡ 2 mod 8.

Established a lower bound for the antipodal number of toroidal grids when rs is odd.

Abstract

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.