A graph G without isolated vertices is a least common multiple of two graphs H1 and H2 if G is a smallest graph, in terms of number of edges, such that there exists a decomposition of G into edge disjoint copies of H1 and there exists a decomposition of G into edge disjoint copies of H2. The concept was introduced by G. Chartrand et al. and they proved that every two nonempty graphs have a least common multiple. Least common multiple of two graphs need not be unique. In fact two graphs can have an arbitrary large number of least common multiples. In this paper graphs that have a unique least common multiple with P3 ∪ K2 are characterized. RESUMENUn grafo G sin vértices aislados es un mínimo común múltiplo de dos grafos H1 y H2 si G es uno de los grafos más pequeños, en términos del número de ejes, tal que existe una descomposición de G en copias de H1 disjuntas por ejes y existe una descomposición de G en copias de H2 disjuntas por ejes. El concepto fue introducido por G. Chartrand et al. donde ellos demostraron que cualquera dos grafos no vaciós tienen un mínimo común múltiplo. El mínimo común múltiplo de dos grafos no es necesariamente único. De hecho, dos grafos pueden tener un número arbitrariamente grande de mínimos comunes múltiplos. En este artículo caracterizamos los grafos que tienen un único mínimo común múltiplo con P3 ∪ K2.

Given an ordered [Formula: see text]-decomposition [Formula: see text] of a connected graph [Formula: see text], the representation of an edge [Formula: see text] with respect to the decomposition [Formula: see text] is the [Formula: see text]-tuple [Formula: see text], where [Formula: see text] represents the distance from [Formula: see text] to [Formula: see text]. A decomposition [Formula: see text] of [Formula: see text] is a resolving decomposition if for every pair of edges [Formula: see text]. The minimum [Formula: see text] for which [Formula: see text] has a resolving [Formula: see text]-decomposition is its decomposition dimension [Formula: see text]. In this paper, upper bounds for the decomposition dimension of some class of graphs are determined.

Let S k+1 denote a star with k edges. Tarsi and Yamamoto et al. have characterized the S k+1 -decomposability of K n the complete graph. In this paper we study the edge decomposition of both K n and the complete bipartite graph K m,n into copies of the union of two edge disjoint stars S p+1 and S q+1 where p ≠q and p,q 2 and obtain the necessary and sufficient conditions for the S p+1 ∪S q+1 -decomposability of K n and K m,n .

A graph [Formula: see text] without isolated vertices is said to be a least common multiple of two graphs [Formula: see text] and [Formula: see text] if [Formula: see text] is a graph of minimum size such that [Formula: see text] is both [Formula: see text] decomposable and [Formula: see text] decomposable. Chartrand et al. proved that every two non-empty graphs [Formula: see text] and [Formula: see text] without isolated vertices have a least common multiple. Size of a least common multiple of [Formula: see text] and [Formula: see text] is denoted by [Formula: see text]. In this paper, least common multiple of some class of graphs are determined.