Keith Edwards

On the Neighbour-Distinguishing Index of a Graph

A proper edge colouring of a graph G is neighbour-distinguishing provided that it distinguishes adjacent vertices by sets of colours of their incident edges. It is proved that for any planar bipartite graph G with Δ(G)≥12 there is a neighbour-distinguishing edge colouring of G using at most Δ(G)+1 colours. Colourings distinguishing pairs of vertices that satisfy other requirements are also considered.

The complexity of harmonious colouring for trees

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. It was shown by Hopcroft and Krishnamoorthy (1983) that the problem of determining the harmonious chromatic number of a graph is NP-hard. We show here that the problem remains hard even when restricted to trees.

Some results on the achromatic number

Let G be a simple graph. The achromatic number ψ(G) is the largest number of colors possible in a proper vertex coloring of G in which each pair of colors is adjacent somewhere in G. For any positive integer m, let q(m) be the largest integer k such that ≤ m. We show that the problem of determining the achromatic number of a tree is NP-hard. We further prove that almost all trees T satisfy ψ (T) = q(m), where m is the number of edges in T. Lastly, for fixed d and ϵ > 0, we show that there is an integer N0 = N0(d, ϵ) such that if G is a graph with maximum degree at most d, and m ≥ N0 edges, then (1 - ϵ)q(m) ≤ ψ (G) ≤ q(m).

Fragmentability of Graphs

We define a parameter which measures the proportion of vertices which must be removed from any graph in a class in order to break the graph up into small (i.e. bounded sized) components. We call this the coefficient of fragmentability of the class. We establish values or bounds for the coefficient for various classes of graphs, particularly graphs of bounded degree. Our main upper bound is proved by establishing an upper bound on the number of vertices which must be removed from a graph of bounded degree in order to leave a planar graph.

New upper bounds on harmonious colorings

We present an improved upper bound on the harmonious chromatic number of an arbitrary graph. We also consider „fragmentable” classes of graphs (an example is the class of planar graphs) that are, roughly speaking, graphs that can be decomposed into bounded-sized components by removing a small proportion of the vertices. We show that for such graphs of bounded degree the harmonious chromatic number is close to the lower bound (2m)1/2, where m is the number of edges.

The computational complexity of cordial and equitable labelling

Abstract A labelling of a simple graph G=(V,E) is an assignment f of integers to the vertices of G. Under such a labelling f, we let Vi denote the set of vertices in G that are labelled i, and let E j ={{u,v}: {u,v}∈E and |f(u)−f(v)|=j}. A k-equitable labelling of a graph G=(V,E) is a labelling f : V→{0,1,…,k−1} such that, for each 0⩽i

THE HARMONIOUS CHROMATIC NUMBER OF BOUNDED DEGREE GRAPHS

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h ( G ) is the least number of colours in such a colouring. Let d be a fixed positive integer, and e>0. We show that there is a natural number M such that if G is any graph with m [ges ] M edges and maximum degree at most d , then the harmonious chromatic number h ( G ) satisfies formula here

The achromatic number of bounded degree trees

Abstract The achromatic number ψ ( G ) of a simple graph G is the largest number of colours possible in a proper vertex colouring of G in which each pair of colours appears on at least one edge. The problem of determining the achromatic number of a tree is known to be NP-hard (Cairnie and Edwards, 1997). In this paper, we present a polynomial-time algorithm for determining the achromatic number of a tree with maximum degree at most d , where d is a fixed positive integer. Prior to giving this algorithm, we show that there is a natural number N ( d ) such that if T is any tree with m ⩾ N ( d ) edges, and maximum degree at most d , then ψ ( T ) is k or k − 1, where k is the largest integer such that k 2 ⩾m .

An Algorithm for Finding Large Induced Planar Subgraphs

This paper presents an efficient algorithm that finds an induced planar subgraph of at least 3n/(d + 1) vertices in a graph of n vertices and maximum degree d. This bound is sharp for d = 3, in the sense that if ɛ > 3/4 then there are graphs of maximum degree 3 with no induced planar subgraph of at least ɛn vertices. Our performance ratios appear to be the best known for small d. For example, when d = 3, our performance ratio of at least 3/4 compares with the ratio 1/2 obtained by Halldorsson and Lau. Our algorithm builds up an induced planar subgraph by iteratively adding a new vertex to it, or swapping a vertex in it with one outside it, in such a way that the procedure is guaranteed to stop, and so as to preserve certain properties that allow its performance to be analysed. This work is related to the authors’ work on fragmentability of graphs.

The Harmonious Chromatic Number of Bounded Degree Trees

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N ( d ) such that if T is any tree with m ≥ N ( d ) edges and maximum degree at most d , then the harmonious chromatic number h ( T ) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d .