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...

The aerodynamics of wind pollination selects for an intimate relation between form and function in anemophilous plants. Inflorescence architecture and floral morphology vary extensively within the Poaceae, but the functional implication of this variation remains largely unknown. Here we quantify associations between floret, culm, and inflorescence characteristics for 25 grass species in Kananas...

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.

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 ...

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. We obtain a new lower bound for the harmonious chromatic number of general graphs, in terms of the independence number of the graph, generalizing results of Moser [2].

Given a graph G and a positive integer p, χp(G) is the minimum number of colours needed to colour the vertices of G so that for any i ≤ p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χp(G) in terms of the k-colouring number colk(G) of G for k = 2p−2. Conversely, for each integer k, we also prove an upper bound for colk(G) in terms of χk+2(G)...

A graph G is chordless if no cycle in G has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree ∆ ≥ 3 has chromatic index ∆ and total chromatic number ∆+1. The proofs are algorithmic in the sense that we ac...

Flowering plants exhibit one of two types of inflorescence architecture: indeterminate, in which the inflorescence grows indefinitely, or determinate, in which a terminal flower is produced. The indeterminate condition is thought to have evolved from the determinate many times, independently. In two mutants in distantly related species, terminal flower 1 in Arabidopsis and centroradialis in Ant...