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arXiv · 2004.05030

Antimagic orientation of lobsters

Yuping Gao·Songling Shan
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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 $\tau:A(D)\rightarrow \{1,2,\cdots,m\}$ such that no two vertices in $D$ have the same vertex-sum under $\tau$, where the vertex-sum of a vertex $u$ in $D$ under $\tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. 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 dense graphs, regular graphs, and trees including caterpillars and $k$-ary trees. In this note, we prove that every lobster admits an antimagic orientation.

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Yuping Gao, Songling Shan. 2020-04-10. Antimagic orientation of lobsters. https://arxiv.org/abs/2004.05030

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