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A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w\left(x\right)={\sum }_{y\in N_G\left(x\right)}\ell \left(y\right)$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover...

It has been shown [3] that any bipartite graph $K_{a,b}$, where a, b are even integers, can be decomposed into closed trails with prescribed even lengths. In this article, we consider the corresponding question for directed bipartite graphs. We show that a complete directed bipartite graph ${}^{\leftrightarrow }{K}_{a,b}$ is decomposable into directed closed trails of even lengths greater than 2, whenever these lengths sum up to the size of the digraph. We use this result to prove that complete bipartite multigraphs can be decomposed...

In the PhD thesis by Burris (Memphis (1993)), a conjecture was made concerning the number of colors c(G) required to edge-color a simple graph G so that no two distinct vertices are incident to the same multiset of colors. We find the exact value of c(G) - the irregular coloring number, and hence verify the conjecture when G is a vertex-disjoint union of paths. We also investigate the point-distinguishing chromatic index, χ₀(G), where sets, instead of multisets, are required to be distinct, and...

Let A = {1, 2, . . . , tm+tn}. We shall say that A has the (m, n, t)-balanced constant-sum-partition property ((m, n, t)-BCSP-property) if there exists a partition of A into 2t pairwise disjoint subsets A1, A2, . . . , At, B1, B2, . . . , Bt such that |Ai| = m and |Bi| = n, and ∑a∈Ai a = ∑b∈Bj b for 1 ≤ i ≤ t and 1 ≤ j ≤ t. In this paper we give sufficient and necessary conditions for a set A to have the (m, n, t)-BCSP-property in the case when m and n are both even. We use this result to show some...

A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ from V to the set {1, . . . , n} such that the weight w(x) = ∑y∈NG(x) ℓ(y) of every vertex x ∈ V is equal to the same element μ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.

Let $\mathrm{Lct}\left(G\right)$ denote the set of all lengths of closed trails that exist in an even graph $G$. A sequence $(t_1,\cdots ,t_p)$ of elements of $\mathrm{Lct}\left(G\right)$ adding up to $\left|E\right(G\left)\right|$ is $G$-realisable provided there is a sequence $(T_1,\cdots ,T_p)$ of pairwise edge-disjoint closed trails in $G$ such that $T_i$ is of length $t_i$ for $i=1,\cdots ,p$. The graph $G$ is arbitrarily decomposable into closed trails if all possible sequences are $G$-realisable. In the paper it is proved that if $a\ge 1$ is an odd integer and $M_{a,a}$ is a perfect matching in $K_{a,a}$, then the graph $K_{a,a}-M_{a,a}$ is arbitrarily decomposable into closed...

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that ${\sum }_{i=1}^k{n}_i=n$, there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set $V_i$ induces a connected subgraph of G on $n_i$ vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.

R. Frucht and J. Gallian (1988) proved that bipartite prisms of order $2n$ have an $\alpha$-labeling, thus they decompose the complete graph $K_{6nx+1}$ for any positive integer $x$. We use a technique called the ${\rho }^{+}$-labeling introduced by S. I. El-Zanati, C. Vanden Eynden, and N. Punnim (2001) to show that also some other families of 3-regular bipartite graphs of order $2n$ called generalized prisms decompose the complete graph $K_{6nx+1}$ for any positive integer $x$.

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a₁,...,aₖ) of positive integers such that a₁+...+aₖ = n there exists a partition (V₁,...,Vₖ) of the vertex set of G such that for each i ∈ 1,...,k, $V_i$ induces a connected subgraph of G on $a_i$ vertices. D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars...

A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ : V → {1, . . . , n} such that the weight of every vertex v, computed as the sum of the labels on the vertices in the open neighborhood of v, is a constant. In this paper, we show that hypercubes with dimension divisible by four are not distance magic. We also provide some positive results by proving necessary and sufficient conditions for the Cartesian product of certain complete multipartite graphs and the cycle on four...