On the \(k\)-distant total labeling of graphs

Abstract

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers. In this paper, two new variations of labeling named $k$-distant edge total labeling and $k$-distant vertex total labeling are introduced. Moreover, the study of two new graph parameters, called $k$-distant edge chromatic number $\left(\gamma_{k d}^{\prime}\right)$ and $k$-distant vertex chromatic number $\left(\gamma_{k d}\right)$ related this labeling are initiated. The $k$-distant vertex total labeling for paths, cycles, complete graphs, stars, bi-stars and friendship graphs are studied and the value of the parameter $\gamma_{k d}$ determined for these graph classes. Then $k$-distant edge total labeling for paths, cycles and stars are studied. Also, an upper bound of $\gamma_{k d}$ and a lower bound of $\gamma_{k d}^{\prime}$ are presented for general graphs.