Antimagic orientation of subdivided caterpillars

Abstract

Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $\tau:A(D)\rightarrow \{1,2,\ldots,m\}$ such that no two vertices in $D$ have the same vertex-sum under $\tau$, where the vertex-sum of a vertex $v$ in $D$ under $\tau$ is the sum of labels of all arcs entering $v$ minus the sum of labels of all arcs leaving $v$. Hefetz, M\"{u}tze and Schwartz [J. Graph Theory, 64: 219-232, 2010] conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of graphs such as regular graphs, graphs with minimum degree at least 33, bipartite graphs with no vertex of degree zero or two, and trees including caterpillars and complete $k$-ary trees. We prove that every subdivided caterpillar admits an antimagic orientation, where a subdivided caterpillar is a subdivision of a caterpillar $T$ such that the edges of $T$ that are not on the central path of $T$ are subdivided the same number of times.