Edge-magic labelings for constellations and armies of caterpillars

TL;DR

This paper proves that symmetric constellations with an odd number of stars and collections of odd numbers of same-type caterpillars have edge-magic or super edge-magic labelings, expanding understanding of graph labelings.

Contribution

It introduces new classes of graphs, such as symmetric constellations and certain caterpillars, that admit edge-magic labelings, providing constructive proofs for these cases.

Findings

Symmetric constellations with an odd number of stars are super edge-magic.

Collections of odd numbers of same-type caterpillars are edge-magic.

The paper extends known classes of graphs with edge-magic labelings.

Abstract

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.