Darryn Bryant

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Repacking in cycle decompositions. A method which takes a decomposition of a graph into edge-disjoint cycles and produces a decomposition of a new graph into edge-disjoint cycles of the same lengths will be discussed. This method, called repacking, underpins much of the recent progress on cycle decomposition problems. Orthogonally resolvable cycle(More)
It is shown that if F 1 , F 2 ,. .. , F t are bipartite 2-regular graphs of order n and α 1 , α 2 ,. .. , α t are non-negative integers such that α 1 +α 2 +· · ·+α t = n−2 2 , α 1 ≥ 3 is odd, and α i is even for i = 2, 3,. .. , t, then there exists a 2-factorisation of K n −I in which there are exactly α i 2-factors isomorphic to F i for i = 1, 2,. .. , t.(More)
In graph cleaning problems, brushes clean a graph by traversing it subject to certain rules. Various problems arise, such as determining the minimum number of brushes that are required to clean the entire graph. This number is called the brushing number. Here, we study a new variant of the brushing problem in which one vertex is cleaned at a time, but more(More)