A characterization of L(2, 1)-labeling number for trees with maximum degree 3

TL;DR

This paper fully characterizes the L(2, 1)-labeling number for trees with maximum degree 3, building on prior work that bounded this number within two values.

Contribution

It provides a complete characterization of the L(2, 1)-labeling number specifically for trees with maximum degree 3, extending previous bounds.

Findings

L(2, 1)-labeling number is either 4 or 5 for these trees

Identifies structural conditions determining the labeling number

Completes the classification for degree 3 trees

Abstract

An L(2, 1)-labeling of a graph is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive numbers differed by at least 2, and vertices at distance 2 are assigned distinct numbers. The L(2, 1)-labeling number is the minimum range of labels over all such labeling. It was shown by Griggs and Yeh [Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5(1992), 586-595] that the L(2, 1)-labeling number of a tree is either \D+ 1 or \D + 2. In this paper, we give a complete characterization of L(2, 1)-labeling number for trees with maximum degree 3.