A graph is (H1,H2)-free for a pair of graphsH1, H2 if it contains no induced subgraph isomorphic toH1 orH2. In 2001, Král’, Kratochvíl, Tuza, and Woeginger initiated a study into the complexity of Colouring for (H1,H2)free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those (H1,H2)-free graphs where H2 is H1, the complement of H1. As thes...

A graph is (H1,H2)-free for a pair of graphsH1, H2 if it contains no induced subgraph isomorphic toH1 orH2. In 2001, Král’, Kratochvíl, Tuza, and Woeginger initiated a study into the complexity of Colouring for (H1,H2)free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those (H1,H2)-free graphs where H2 is H1, the complement of H1. As thes...

For \(k\ge 1\), a k-colouring c of G is mapping from V(G) to \(\{1,2,\ldots ,k\}\) such that \(c(u)\ne c(v)\) for any two non-adjacent vertices u and v. The k -Colouring problem decide if graph has k-colouring. family graphs \(\mathcal{H}\), \(\mathcal{H}\)-free does not contain \(\mathcal{H}\) as an induced subgraph. Let \(C_s\) be the s-vertex cycle. In previous work (MFCS 2019) we examined e...

Perfect colouring of isonemal fabrics by thin and thick striping of warp and weft with more than two colours is introduced. Conditions that prevent perfect colouring by striping are derived, and it is shown that avoiding the preventing conditions is sufficient to allow perfect colouring. Examples of thick striping in all possible species are

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 define k-diverse colouring of a graph to be a proper vertex colouring in which every vertex x, sees min{k, d(x)} different colours in its neighbors. We show that for given k there is an f(k) for which every planar graph admits a k-diverse colouring using at most f(k) colours. Then using this colouring we obtain a K5-free graph H for which every planar graph admits a homomorphism to it, thus ...

We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordere...