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null (Ed.)A longstanding conjecture by Kotzig, Ringel, and Rosa states that every tree admits a graceful labeling. That is, for any tree $T$ with $n$~edges, it is conjectured that there exists a labeling $f\colon V(T) \to \{0,1,\ldots,n\}$ such that the set of induced edge labels $\bigl\{ f(u)f(v) : \{u,v\}\in E(T) \bigr\}$ is exactly $\{1,2,\ldots,n\}$. We extend this concept to allow for multigraphs with edge multiplicity at most~$2$. A \emph{2fold graceful labeling} of a graph (or multigraph) $G$ with $n$~edges is a onetoone function $f\colon V(G) \to \{0,1,\ldots,n\}$ such that the multiset of induced edge labels is comprised of two copies of each element in $\bigl\{ 1,2,\ldots, \lfloor n/2 \rfloor \bigr\}$ and, if $n$ is odd, one copy of $\bigl\{ \lceil n/2 \rceil \bigr\}$. When $n$ is even, this concept is similar to that of 2equitable labelings which were introduced by Bloom and have been studied for several classes of graphs. We show that caterpillars, cycles of length $n \not\equiv 1 \pmod{4}$, and complete bipartite graphs admit 2fold graceful labelings. We also show that under certain conditions, the join of a tree and an empty graph (i.e., a graph with vertices but no edges) is $2$fold graceful.more » « less

Bermudez, H ; Bunge, R ; Cornelius III, E ; ElZanati, S ; Mamboleo, W ; Nguyen, N ; Roberts, D. ( , Journal of Combinatorial Mathematics and Combinatorial Computing)null (Ed.)Let $G$ be one of the two multigraphs obtained from $K_4e$ by replacing two edges with a doubleedge while maintaining a minimum degree of~2. We find necessary and sufficient conditions on $n$ and $\lambda$ for the existence of a $G$decomposition of $^{\lambda}K_n$.more » « less

Bunge, R ; ElZanati, S ; Hawken, K ; Ramirez, E ; Roberts, D ; RodriguezGuzman, E ; Williams, J. ( , Journal of Combinatorial Mathematics and Combinatorial Computing (ISSN: 08353026))null (Ed.)Consider the multigraph obtained by adding a double edge to $K_4e$. Now, let $D$ be a directed graph obtained by orientating the edges of that multigraph. We establish necessary and sufficient conditions on $n$ for the existence of a $(K^{*}_{n},D)$design for four such orientations.more » « less