Henry Liu

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Suppose the edges of the complete graph on n vertices, E(K n), are coloured using r colours; how large a k-connected sub-graph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s = 1 was considered in [3]. Here we shall consider the case s 2, proving in particular that when s = 2 and r + 1 is(More)
We consider the following question of Bollobás: given an r-colouring of E(K n), how large a k-connected subgraph can we find using at most s colours? We provide a partial solution to this problem when s = 1 (and n is not too small), showing that when r = 2 the answer is n − 2k + 2, when r = 3 the answer is ⌊ n−k 2 ⌋ + 1 or ⌈ n−k 2 ⌉ + 1, and when r − 1 is a(More)
Although measuring and archiving freeway traffic performance using commonly available loop detector data has become a norm for many transportation agencies, similar approaches for urban arterials do not exist. In practice, operational data from traffic signal systems are neither stored nor analyzed, which prevents the proactive management of arterial(More)
The balanced decomposition number f (G) of a graph G was introduced by Fujita and Nakamigawa [Discr. A balanced colouring of a graph G is a colouring of some of the ver-tices of G with two colours, such that there is the same number of vertices in each colour. Then, f (G) is the minimum integer s with the following property: For any balanced colouring of G,(More)
A balanced colouring of a graph G is a colouring of some of the vertices of G with two colours, say red and blue, such that there is the same number of vertices in each colour. The balanced decomposition number f (G) of G is the minimum integer s with the following property: For any balanced colouring of G, there is a partition V (G) = V 1 ˙ ∪ · · · ˙ ∪ V r(More)
Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let φ k (n, H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic(More)