Anti-$k$-labeling of graphs

TL;DR

This paper introduces the anti-$k$-labeling problem for graphs, aiming to minimize neighboring node similarity, with applications in frequency assignment, and provides bounds and complexity results for various graph classes.

Contribution

It formalizes the anti-$k$-labeling problem, derives a formula for the anti-$k$-labeling number, and establishes NP-hardness along with bounds for specific graph types.

Findings

The anti-$k$-labeling number is given by a specific formula involving chromatic number.

Determined that computing $mc_k(G)$ is NP-hard.

Provided lower bounds for trees, grids, and $n$-cubes.

Abstract

It is well known that the labeling problems of graphs arise in many (but not limited to) networking and telecommunication contexts. In this paper we introduce the anti-$k$-labeling problem of graphs which we seek to minimize the similarity (or distance) of neighboring nodes. For example, in the fundamental frequency assignment problem in wireless networks where each node is assigned a frequency, it is usually desirable to limit or minimize the frequency gap between neighboring nodes so as to limit interference. Let $k\geq1$ be an integer and $\psi$ is a labeling function (anti-$k$-labeling) from $V(G)$ to $\{1,2,\cdots,k\}$ for a graph $G$. A {\em no-hole anti-$k$-labeling} is an anti-$k$-labeling using all labels between 1 and $k$. We define $w_{\psi}(e)=|\psi(u)-\psi(v)|$ for an edge $e=uv$ and $w_{\psi}(G)=\min\{w_{\psi}(e):e\in E(G)\}$ for an anti-$k$-labeling $\psi$ of the graph $G$. {\em The anti-$k$-labeling number} of a graph $G$, $mc_k(G)$ is $\max\{w_{\psi}(G): \psi\}$. In this paper, we first show that $mc_k(G)=\lfloor \frac{k-1}{\chi-1}\rfloor$, and the problem that determines $mc_k(G)$ of graphs is NP-hard. We mainly obtain the lower bounds on no-hole anti-$n$-labeling number for trees, grids and $n$-cubes.