We prove that all Paley graphs can be decomposed into Hamilton cycles. This paper is dedicated to Gert Sabidussi in celebration of his 80th birthday.
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)
It has been conjectured that any partial triple system of order u and index λ can be embedded in a triple system of order v and index λ whenever v ≥ 2u + 1, λ(v − 1) is even and λ v 2 ≡ 0 (mod 3). This conjecture is known to hold for λ = 1 and for all even λ ≥ 2. Here the conjecture is proven for all remaining values of λ when u ≥ 28.
It is shown that if G is any bipartite 2-regular graph of order at most n 2 or at least n − 2, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order n into a perfect matching and edge-disjoint copies of G.
We prove that if H = (V (H), E(H)) is a hypergraph, γ is an edge colouring of H, and S ⊆ V (H) such that any permutation of S is an automorphism of H, then there exists a permutation π of E(H) such that |π(E)| = |E| and π(E) \ S = E \ S for each E ∈ E(H), and such that the edge colouring γ of H given by γ (E) = γ(π −1 (E)) for each E ∈ E(H) is almost… (More)
It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.