Edge-magic labelings for constellations and armies of caterpillars

Abstract

Let $G=(V,E)$ be an $n$-vertex graph with $m$ edges. A function $f:V\cup E\to \{1,\dots ,n+m\}$ is an edge-magic labeling of $G$ if $f$ is bijective and, for some integer $k$, we have $f(u)+f(v)+f(uv)=k$ for every edge $uv\in E$. Furthermore, if $f(V)=\{1,\dots ,n\}$, then we say that $f$ is a super edge-magic labeling. A constellation, which is a collection of stars, is symmetric if the number of stars of each size is even except for at most one size. We prove that every symmetric constellation with an odd number of stars admits a super edge-magic labeling. We say that a caterpillar is of type $(r,s)$ if $r$ and $s$ are the sizes of its parts, where $r\le s$. We also prove that every collection with an odd number of same-type caterpillars admits an edge-magic labeling.