Radio Labelings of Distance Graphs

TL;DR

This paper investigates radio labelings of distance graphs, focusing on assigning non-negative integers to vertices such that the label differences meet specific distance-dependent constraints, with implications for graph labeling theory.

Contribution

The paper introduces new results on radio labelings of distance graphs with integer vertices and adjacency defined by a set D, expanding understanding of graph labeling constraints.

Findings

Derived bounds for radio labelings of distance graphs.

Characterized optimal labelings for specific distance sets.

Extended previous results to broader classes of distance graphs.

Abstract

A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.