Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let φk(n,H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic H-decomposition with at most φ parts. Here we stud...

ABSTRACT: A difference labeling of a graph G is realized by assigning distinct integer values to its vertices and then associating with each edge the absolute difference of those values assigned to its end vertices. A decomposition of labeled graph into parts, each part containing the edge having a commonweight is called a common – weight decomposition. In this paper we investigate the existenc...

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted L inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Ma...

We study the decomposition of multigraphs with a constant edge multiplicity into copies of a fixed starH =K1,t :We present necessary and sufficient conditions for such a decomposition to exist where t = 2 and prove NP-completeness of the corresponding decision problem for any t 3.We also prove NP-completeness when the edge multiplicity function is not restricted either on the input G or on the ...

A P -decomposition of a graph G is a set of pairwise edge-disjoint paths of G with edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an -regular graph to admit a P -decomposition, for odd . Let g, and m be positive integers wi...

The Tree Decomposition Conjecture by Barát and Thomassen states that for every tree T there exists a natural number k(T ) such that the following holds: If G is a k(T )-edge-connected simple graph with size divisible by the size of T , then G can be edge-decomposed into subgraphs isomorphic to T . So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove ...

Let G be a connected graph on n vertices, and let ; ; and be edge-disjoint cycles in G such that (i) ; (resp. ;) are vertex-disjoint and (ii) jj + jj = jj+ jj = n, where jj denotes the length of. We say that ; ; and yield two edge-disjoint hamiltonian cycles by edge exchanges if the four cycles respectively contain edges e; f; g and h such that each of (? feg) S (? ffg) S fg; hg and (? fgg) S (...