Total difference chromatic numbers of graphs

Abstract

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 $\left\{1,2,\dots ,k\right\}$ that is a $k$-total labeling of $G$ and for which $f\left(\left\{u,v\right\}\right)=|f\left(u\right)-f\left(v\right)|$ for any two adjacent vertices $u$ and $v$ of $G$ with incident edge $\left\{u,v\right\}$. The least positive integer $k$ for which $G$ has a $k$-total difference labeling is its total difference chromatic number, ${\chi }_{td}\left(G\right)$. 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.