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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)
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)
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)
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)
Epinephrine is often used for the treatment of patients with heart failure, low cardiac output and cardiac arrest. It can acutely improve hemodynamic parameters; however, it does not seem to improve longer term clinical outcomes. Therefore, we hypothesized that epinephrine may induce unfavorable changes in gene expression of cardiomyocyte. Thus, we(More)
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