Total Difference Labeling of Regular Infinite Graphs

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.