With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishin...

A k-frugal colouring of a graph G is a proper colouring of the vertices of G such that no colour appears more than k times in the neighbourhood of a vertex. This type of colouring was introduced by Hind, Molloy and Reed in 1997. In this paper, we study the frugal chromatic number of planar graphs, planar graphs with large girth, and outerplanar graphs, and relate this parameter with several wel...

statement of problem: acrylic teeth of dentures are prone to discoloration due to frequent exposure to different colouring solutions in the oral environment. thus, in order to maintain their aesthetic appearance, they must possess high colour stability while exposed to colorant solutions. objectives: to compare the colour stability of three different brands of acrylic teeth following immersion ...

A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2, . . . , v2t such that vi and vt+i receive the same colour for all i = 1, 2, . . . , t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous boun...

Grass species differ in many aspects of inflorescence architecture, but in most cases the genetic basis of the morphological difference is unknown. To investigate the genes underlying the morphology in one such instance, we undertook a developmental and QTL analysis of inflorescence differences between the cereal grain foxtail millet and its presumed progenitor green millet. Inflorescence diffe...

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a nonlocally finite graph for w...

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is NP-complete. We show that for every choice of positive integers r, k there exists a k-strong bipartite tournament which needs at least r colours in...

The effect of inflorescence number and distribution along the shoot on the level of fruit-set was studied using `ON' year olive trees with a high level of floral differentiation. Reduced levels and different inflorescence distribution patterns were created artificially by hand inflorescence thinning. In most cases, removal of up to 50% of the inflorescences had either no effect on the total amo...

Interactions between flowering time genes were examined in a doubled haploid barley (Hordeum vulgare) population segregating for H. vulgare VERNALIZATION1 (HvVRN1), HvVRN2, and PHOTOPERIOD1 (PPD-H1). A deletion allele of HvVRN2 was associated with rapid inflorescence initiation and early flowering, but only in lines with an active allele of PPD-H1. In these lines, the floral promoter FLOWERING ...