The study of graph labellings has focused on finding classes of graphs which admit a particular type of labelling. Here we consider variations of the well-known edge-magic and vertex-magic total labellings for which all graphs admit such a labelling. In particular, we consider two types of injections of the vertices and edges of a graph with positive integers: (1) for every edge the sum of its ...

Let G = (V,E) be a graph of order n. A closed distance magic labeling of G is a bijection l : V (G) → {1, . . . , n} for which there exists a positive integer k such that ∑ x∈N [v] l(x) = k for all v ∈ V , where N [v] is the closed neighborhood of v. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic la...

A graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E1, ..., Ek of E(G) such that each G[Ei] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przyby lo and Woźniak that they decompose i...

A walk on an edge-colored graph G is a path in G which contains each edge at least once, and the trace of a walk w is the sequence of edge-colors seen in w. We investigate the walk realizability problem, which is to decide, given an edge-colored graph G and a string x of colors, if there is a walk on G with trace x. We de ne a method of decomposing graphs into linear chains, and show that the n...

He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching. c © 2008 Elsevier B.V. All rights reserved.

The proof of the following theorem is the main result of this paper: If G is a 2k-regular graph that has a perfect 1-factorisation, then the line graph, L(G), of G is Hamilton decomposable. Consideration is given to Hamilton decompositions of L(K 2k ? F).

In this paper we show that the complete graph K 12 is not decomposable into three factors with diameter two, which completes the solution of the problem of decomposition of complete graph into three factors from which the rst one has diameter two and the other factors have nite diameters.