The facial parity edge colouring of a connected bridgeless plane graph is such an edge colouring in which no two face-adjacent edges (consecutive edges of a facial walk of some face) receive the same colour, in addition, for each face α and each colour c, either no edge or an odd number of edges incident with α is coloured with c. From the Vizing’s theorem it follows that every 3-connected plan...

This paper discusses the game colouring number of partial k-trees and planar graphs. Let colg(PT k) and colg(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this paper, we prove that colg(PT k) = 3k + 2 and colg(P) ≥ 11. We also prove that the game colouring number colg(G) of a graph is a monotone parameter,...

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of three edges uses three colours. An induced matching of a graph G is a subset I of edges of G such that the graph induced by the endpoints of I is a matching. In this paper, we prove the NP-completeness of strong 4, 5, and 6-edge-colouring and maximum induced matching in some subclasses of subcubic triangle-f...

We introduce the notion of (A,B)-colouring of a graph: For given vertex sets A, B, this is a colouring of the vertices in B so that both adjacent vertices and vertices with a common neighbour in A receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of plane graphs. We prove a general result which implies asymptotic versions of...

Introduction Few things influence us as much as colour and few things engage and interest people as much as colour. Colour plays an essential role in the environment in which we live, whether it is at our place of work, in the public environment or in our home. To choose the right colour is necessary to attain the desired effect in colouring and design, both in the colouring of our environment ...

relationship between grain yield and its component traits can improve the efficiency of breeding programs by determining appropriate selection criteria. an investigation was carried out on barnyard millet (echinochloa spp.) global germplasm collection to investigate the association among yield components and their direct and indirect effects on the grain yield of barnyard millet. the experiment...

An edge colouring of a multigraph can be thought of as a partition of the edges into matchings (a matching meets each vertex at most once). Analogously, an edge cover colouring is a partition of the edges into edge covers (an edge cover meets each vertex at least once). We aim to determine a tight lower bound on the maximum number of parts in an edge cover colouring as a function of the minimum...

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...