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

TL;DR

This work addresses the antipodal coloring problem in radio $k$-coloring of graphs by introducing a practical sufficient criterion for minimal antipodal colorings. It then applies the criterion to two important graph families: the generalized Petersen graphs $GP(n,1)$ and toroidal grids $T_{r,s}=C_r \square C_s$, yielding explicit formulas for the antipodal number $ac(G)$ in most cases and establishing an upper bound in a notable exception for $GP(n,1)$. For toroidal grids, the authors provide a comprehensive set of exact values for $ac(T_{r,s})$ across multiple residue classes modulo $4$ and a lower bound when $rs$ is odd. The results advance exact antipodal coloring numbers in key network-graph contexts and offer a constructive method to verify minimal antipodal colorings.

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.

Figures (1)

• Figure 1: Ordering of vertices and mapping $g$ for $GP(8,1)$

Theorems & Definitions (26)

• Lemma 1.1
• Remark 1.1
• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Remark 3.1
• Theorem 3.1
• Remark 4.1
• Corollary 4.1