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Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph 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)
The balanced decomposition number f(G) of a graph G was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339-3344]. A balanced colouring of a graph G is a colouring of some of the vertices 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(More)
Let G be a graph whose edges are colored with k colors, and H=(H1,⋯,Hk) be a k-tuple of graphs. 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 copy of Hi in color i, for some 1≤i≤k. Let φk(n,H) be the smallest number ϕ, such that, for every order-n graph and every(More)
Let k be a positive integer andG be a k-connected graph. An edge-coloured path is rainbow if its edges have distinct colours. The rainbow k-connection number of G, denoted by rck(G), is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function(More)
We consider the following question of Bollobás: given an r-colouring of E(Kn), 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 prime(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 φk(n,H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic(More)
Let k be a positive integer and G be a k-connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k-connection number rck(G) of G. An edge-coloured path is rainbow if its edges have distinct colours. Then, rck(G) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k(More)
In this paper, we present an adaptive signal control scheme to prevent intersection traffic blockage resulted from vehicle queue spillover. A method to identify vehicle queue spillover condition through simplified shockwave analysis is developed. Instead of measuring the vehicle queue length or locating the end of queue directly, this method relies on the(More)