Total Difference Chromatic Numbers of Graphs

Inspired by graceful labelings and total labelings of graphs, we introduce the idea of total difference labelings. A \(k\)-total labeling of a graph \(G\) is an assignment of \(k\) distinct labels to the edges and vertices of a graph so that adjacent vertices, incident edges, and an edge and its incident vertices receive different labels. A \(k\)-total difference labeling of a graph \(G\) is a function \(f\) from the set of edges and vertices of \(G\) to the set \(\{1,2,\ldots,k\}\), that is a \(k\)-total labeling of \(G\) and for which \(f(\{u,v\})=|f(u)-f(v)|\) for any two adjacent vertices \(u\) and \(v\) of \(G\) with incident edge \(\{u,v\}\). The least positive integer \(k\) for which \(G\) has a \(k\)-total difference labeling is its total difference chromatic number, \(\chi_{td}(G)\). We determine the total difference chromatic number of paths, cycles, stars, wheels, gears and helms. We also provide bounds for total difference chromatic numbers of caterpillars, lobsters, and general trees.