On the Proper Arc Labeling of Directed Graphs

Abstract

An arc labeling $\ell$ of a directed graph G with positive integers is proper if for any two adjacent vertices v, u, we have $S_{\ell }(v) \ne S_{\ell }(u)$, where $S_{\ell }(v)$ denotes the sum of labels of the arcs with head v. The proper arc labeling number of a directed graph G, denoted by $\overrightarrow{{\chi }_p}(G)$, is $\min _{\ell \in \Gamma }\max _{v\in V(G)} S_{\ell }(v)$, where $\Gamma$ is the set of proper arc labelings of G. A proper arc labeling $\ell$ of G is optimal if $\max _{v\in V(G)} S_{\ell }(v)= \overrightarrow{{\chi }_p}(G)$. In this work we study the proper arc labeling number of graphs and present some lower and upper bounds. We show that every directed graph G has a (not necessarily optimal) proper arc labeling from $\{1,2,3\}$. So, $\Delta ^{-}(G) \le \overrightarrow{{\chi }_p}(G) \le 3\Delta ^{-}(G)$, where $\Delta ^{-}(G)$ is the maximum incoming degree of vertices in G. Next, we prove that every graph G has an optimal proper arc labeling from $\{1,2,\ldots , r+1\}$, where $r=\max \{\lceil \frac{d^+_G(v)}{d^-_G(v)}\rceil +1 : v\in V(G) \}$. Among other results, we show that if G is a bipartite graph, then $\Delta ^{-}(G) \le \overrightarrow{{\chi }_p}(G) \le \Delta ^{-}(G)+1$. Furthermore, we show that for each constant number c, there is a directed tree T such that it does not have any optimal proper arc labeling from $\{1,2,\ldots , c\}$. Next, we show that there is a polynomial time algorithm to find the proper arc labeling number of directed trees. On the other hand, we show that computing the proper arc labeling number of planar directed graphs is NP-hard.