An undirected simple graphG is locally irregular if adjacent vertices ofG have different degrees. An edge-colouring φ ofG is locally irregular if each colour class of φ induces a locally irregular subgraph of G. The irregular chromatic index χirr(G) of G is the least number of colours used by a locally irregular edge-colouring of G (if any). We show that the problem of determining the irregular...

In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1, 2, 3, . . . , 2n − 1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|/4 for any non-empty subset R ⊆ C, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors, namely, any color appe...

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with...

This paper is an expository piece on edge-chromatic graph theory. The central theorem in this subject is that of Vizing. We shall then explore the properties of graphs where Vizing’s upper bound on the chromatic index is tight, and graphs where the lower bound is tight. Finally, we will look at a few generalizations of Vizing’s Theorem, as well as some related conjectures.

We give upper bounds for the generalised acyclic chromatic number and generalised acyclic edge chromatic number of graphs with maximum degree d, as a function of d. We also produce examples of graphs where these bounds are of the correct order.

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to...

The harmonious coloring of an undirected simple graph is a vertex coloring such that adjacent vertices are assigned different colors and each pair of colors appears together on at most one edge. The harmonious chromatic number of a graph is the least number of colors used in such a coloring. The harmonious chromatic number of a path is known, whereas the problem to find the harmonious chromatic...

The strong chromatic index of a graph G is the smallest integer k such that the edge set E( G) can be partitioned into k induced subgraphs of G which form matchings. In this paper we consider the behavior of the strong chromatic index of a sparse random graph K (n, p), where p = p(n) = 0(1).