Total Difference Labeling of Regular Infinite Graphs

TL;DR

This paper investigates total difference labelings of graphs, providing improved bounds for complete graphs and extending results to infinite regular graphs, advancing understanding of graph labelings.

Contribution

It improves the upper bound on the total difference labeling number for complete graphs and extends theoretical results to infinite regular graphs.

Findings

Improved upper bound on $oldsymbol{ ext{chi}_{td}(K_n)}$

Extended total difference labeling results to infinite regular graphs

Enhanced understanding of graph labeling bounds

Abstract

Given a graph $G$, a \textit{$k$-total difference labeling} of the graph is a total labeling $f$ from the set of edges and vertices to the set $\{1, 2, \cdots k\}$ satisfying that for any edge $\{u,v\}$, $f(\{u,v\})=|f(u)-f(v)|$. If $G$ is a graph, then $\chi_{td}(G)$ is the minimum $k$ such that there is a $k$-total difference labeling of $G$ in which no two adjacent labels are identical. We extend prior work on total difference labeling by improving the upper bound on $\chi_{td}(K_n)$ and also by proving results concerning infinite regular graphs.