星座图与毛虫军团的边魔幻标号
Edge-magic labelings for constellations and armies of caterpillars
AI总结：
该研究针对图论中的边魔幻标号问题，证明了含奇数个星图的对称星座图存在超级边魔幻标号，且奇数个同型毛虫图的集合存在边魔幻标号，拓展了两类图的标号理论结果。
摘要：
设$G=(V,E)$为具有$n$个顶点、$m$条边的图。若函数$f : V \cup E \rightarrow \{1, \ldots, n+m\}$是双射，且存在整数$k$使得对每条边$uv \in E$都有$f(u)+f(v)+f(uv) = k$，则称$f$为$G$的边魔幻标号。此外，若$f(V) = \{1, \ldots, n\}$，则称$f$为超级边魔幻标号。星座图是若干星图的集合，若除至多一种规模外，每种规模的星图数量均为偶数，则称其为对称星座图。我们证明：每个具有奇数个星图的对称星座图都存在超级边魔幻标号。若毛虫图的两部分规模为$r$和$s$（$r \leq s$），则称其为$(r,s)$型毛虫图。我们还证明：每个由奇数个同型毛虫图组成的集合都存在边魔幻标号。
英文摘要：
Let $G=(V,E)$ be an $n$-vertex graph with $m$ edges. A function $f : V \cup E \rightarrow \{1, \ldots, n+m\}$ 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 $f(V) = \{1, \ldots, n\}$, 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 $r \leq s$. We also prove that every collection with an odd number of same-type caterpillars admits an edge-magic labeling.
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