© 2018 Wiley Periodicals, Inc. We show that any complete k-partite graph G on n vertices, with k≥3, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours, given that ...

Let G be a graph on n vertices with maximum degree ?. We use the Lovasz local lemma to show the following two results about colourings ? of the edges of the complete graph Kn. If for each vertex v of Kn the colouring ? ...

The distance-two labelling problem of graphs was proposed by Griggs and Roberts in 1988, and it is a variation of the frequency assignment problem introduced by Hale in 1980. An L(2, 1)-labelling of a graph G is an assignment ...

Given a c-edge-coloured multigraph, where c is a positive integer, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we ...

The first part of this thesis concerns monochromatic cycle partitions. We make the following three contributions. Our first result is that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 ...