A harmonious colouring of a simple graph G is a colouring of the vertices such that adjacent vertices receive distinct colours and 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. We improve an upper bound on h(G) due to Lee and Mitchem, and give upper bounds for related quantities.

We present the first linear time algorithm for d-list colouring of a graph—i.e. a proper colouring of each vertex v by colours coming from lists L(v) of sizes at least deg(v). Previously, procedures with such complexity were only known for ∆-list colouring, where for each vertex v one has |L(v)| ≥ ∆, the maximum of the vertex degrees. An implementation of the procedure is available.

A vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. in [9] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of pla...

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible. Consider a graph G where each edge of G is assigned a colour from {1, ..., k} (this is not necessarily a proper edge colouring). A k-adapted colouring is an assignment of colours from {1, ..., k} to the vertices of G such that there is no edge with th...

Approximate random k-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdős-Rényi random graph G(n, d/n), where d is a sufficiently large constant. We propose a n...

A colouring of the edges of an n×n grid is said to be reconstructible if the colouring is uniquely determined by the multiset of its n tiles, where the tile corresponding to a vertex of the grid specifies the colours of the edges incident to that vertex in some fixed order. In 2015, Mossel and Ross asked the following question: if the edges of an n× n grid are coloured independently and uniform...

3-list colouring is an NP-complete decision problem. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving 3-list colouring on permutation graphs.

Approximate random k-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdős-Rényi random graph G(n, d/n), where d is a sufficiently large constant. We propose a n...

A natural way to colour the vertices of a graph is: (i) to impose a linear order < on the vertices, and (ii) to scan the vertices in this order, assigning to each vertex c(,j) the smallest positive integer assigned to no neighbour v(k) of o(j) with z>(k) < t:(,j). This heuristic algorithm is called the greedy colouring algorithm, or the sequential colouring algorithm. One may ask the following ...