Total difference chromatic numbers of regular infinite graphs

Given a graph \( G \), a \( k \)-total difference labeling of the graph is a total labeling \( f \) from the set of edges and vertices to the set \(\{1, 2, \ldots, k\}\) satisfying \( f(\{u,v\}) = |f(u) - f(v)| \) for any edge \(\{u,v\}\). If \( G \) is a graph, then \( \chi_{\text{td}}(G) \) 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 \( \chi_{\text{td}}(K_n) \) and also by proving results concerning infinite regular graphs.