(2,1)-Total labeling of planar graphs with large maximum degree

TL;DR

This paper proves that for planar graphs with maximum degree at least 12, the (2,1)-total labeling number is at most two more than the maximum degree, advancing understanding of graph labelings.

Contribution

It establishes an upper bound of +2 for the (2,1)-total labeling number of planar graphs with , which was previously unknown.

Findings

+2 upper bound for  total labeling

Applicable to planar graphs with 

Extends previous bounds for graph labelings

Abstract

The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $\Delta\geq12$ and $d=2$, the (2,1)-total labelling number $\lambda_2^T(G)$ is at most $\Delta+2$.