Dispersed graph labellings

TL;DR

This paper investigates the maximum dispersion in graph labellings, providing bounds, exact values for specific graph classes, and new construction methods, despite the NP-hardness of the problem.

Contribution

It introduces bounds, exact values for key graphs, and novel construction techniques for dispersed graph labellings, advancing understanding of this NP-hard problem.

Findings

Exact $DL(G)$ for cycles, paths, grids, hypercubes, and binary trees.

Established degree-based bounds and product constructions.

Proved NP-hardness of computing $DL(G)$.

Abstract

A $k$-dispersed labelling of a graph $G$ on $n$ vertices is a labelling of the vertices of $G$ by the integers $1, \dots , n$ such that $d(i,i+1) \geq k$ for $1 \leq i \leq n-1$. $DL(G)$ denotes the maximum value of $k$ such that $G$ has a $k$-dispersed labelling. In this paper, we study upper and lower bounds on $DL(G)$. Computing $DL(G)$ is NP-hard. However, we determine the exact values of $DL(G)$ for cycles, paths, grids, hypercubes and complete binary trees. We also give a product construction and we prove a degree-based bound.