\((2,1)\)-Total Labeling Of Planar Graphs with Large Girth And Low Maximum Degree

Abstract

A \((2,1)\)-total labeling of a graph \(G\) is a labeling of vertices and edges, such that:(1) any two adjacent vertices of \(G\) receive distinct integers,(2) any two adjacent edges receive distinct integers, and (3) a vertex and its incident edges receive integers that differ by at least 2 in absolute value.The span of a \((2,1)\)-total labeling is the difference between the maximum label and the minimum label.We note the minimum span \(\lambda_2^T(G)\).In this paper, we prove that if \(G\) is a planar graph with \(\Delta \leq 3\) and girth \(g \geq 18\), then \(\lambda_2^T(G) \leq 5\). If \(G\) is a planar graph with \(\Delta \leq 4\) and girth \(g \geq 12\), then \(\lambda_2^T(G) \leq 7\).