Graph Labeling with Distance Conditions and the Delta Squared Conjecture

TL;DR

This paper establishes bounds on the L(2,1)-labeling number of graphs based on their size and maximum degree, and identifies an infinite class of graphs with extremal labeling numbers.

Contribution

It provides new bounds for the L(2,1)-labeling number and characterizes an infinite class of graphs with maximal labeling numbers relative to their degrees.

Findings

Derived bounds on the L(2,1)-labeling number based on graph order and maximum degree.

Identified an infinite class of graphs with the highest known L(2,1)-labeling numbers.

Characterized extremal graphs with respect to the L(2,1)-labeling problem.

Abstract

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their maximum degrees of any known infinite class of graphs.