A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(G) ≤ ∆∗ + 2 for any 3-connected plane graph G with maximum face degree ∆∗. It is known that th...

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χa(G) is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and ∆(G) is large enough then χa(G) = ∆(G). We settle this conjecture for planar gr...

A system of linear equations L over Open image in new window is common if the number monochromatic solutions to any two-colouring asymptotically at least a random . The line research on systems was recently initiated by Saad and Wolf. They were motivated existing results for specific (such as Schur triples arithmetic progressions), well extensive Sidorenko graphs. Building earlier work, Fox, Ph...

To investigate drawing development, kindergarten (N = 213; age range 2;0–3;9and Grade 1 (N = 183; age range 6;11–8;9) children performed Moore’s (1986) colouring task. It was found that young children’s drawings of a cube represent generalizations rather than particular models. An intermediate stage of differentiation between scribbles and representational drawings, closed forms, was identified...

The central problem of the total-colourings is the Total-Colouring Conjecture, which asserts that every graph of maximum degree ∆ admits a (∆ + 2)-total-colouring. Similarly to edge-colourings—with Vizing’s edge-colouring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if ∆ ≥ 10 then every plane graph of maximum degree ∆ is...

In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph wi...

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χa(G), is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and ∆(G) is large enough then χa(G) = ∆(G). We settle this conjecture for planar g...

The Sum Colouring Problem is an NP-hard problem derived from the well-known graph colouring problem. It consists in finding a proper colouring which minimizes the sum of the assigned colours rather than the number of those colours. This problem often arises in scheduling and resource allocation. In this paper, we conduct an in-depth evaluation of ILP and CP’s capabilities to solve this problem,...

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χa(G), is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and ∆(G) is large enough then χa(G) = ∆(G). We settle this conjecture for planar g...

We consider edge colourings of K n – the complete r-uniform hypergraph on n vertices. Our main question is: how ‘colourful’ can such a colouring be if we restrict the number of colours locally? The local restriction is formulated as follows: for a fixed hypergraph H and an integer k we call a colouring (H, k)-local, if every copy of H in the complete hypergraph K n picks up at most k different ...