45 research outputs found
2-factors with k cycles in Hamiltonian graphs
A well known generalisation of Dirac's theorem states that if a graph on
vertices has minimum degree at least then contains a
-factor consisting of exactly cycles. This is easily seen to be tight in
terms of the bound on the minimum degree. However, if one assumes in addition
that is Hamiltonian it has been conjectured that the bound on the minimum
degree may be relaxed. This was indeed shown to be true by S\'ark\"ozy. In
subsequent papers, the minimum degree bound has been improved, most recently to
by DeBiasio, Ferrara, and Morris. On the other hand no
lower bounds close to this are known, and all papers on this topic ask whether
the minimum degree needs to be linear. We answer this question, by showing that
the required minimum degree for large Hamiltonian graphs to have a -factor
consisting of a fixed number of cycles is sublinear in Comment: 13 pages, 6 picture
A counterexample to Stein's Equi-n-square conjecture
In 1975 Stein conjectured that in every n × n array filled with the numbers 1,...,n with every number occuring exactly n times, there is a partial transversal of size n − 1. In this note we show that this conjecture is false by constructing such arrays without partial transverals of size n−(1/42)ln(n)
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different
colours. The study of rainbow decompositions has a long history, going back to the work of
Euler on Latin squares. In this paper we discuss three problems about decomposing complete
graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and
the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the
best known bounds on all these conjectures. Using our method we also show that every properly
(n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on
the Brualdi-Hollingsworth Conjecture
C4-free subgraphs with large average degree
Motivated by a longstanding conjecture of Thomassen, we study how large the
average degree of a graph needs to be to imply that it contains a -free
subgraph with average degree at least . K\"uhn and Osthus showed that an
average degree bound which is double exponential in t is sufficient. We give a
short proof of this bound, before reducing it to a single exponential. That is,
we show that any graph with average degree at least (for
some constant ) contains a -free subgraph with average degree at
least . Finally, we give a construction which improves the lower bound for
this problem, showing that this initial average degree must be at least
A proof of Ringel's Conjecture
A typical decomposition question asks whether the edges of some graph can
be partitioned into disjoint copies of another graph . One of the oldest and
best known conjectures in this area, posed by Ringel in 1963, concerns the
decomposition of complete graphs into edge-disjoint copies of a tree. It says
that any tree with edges packs times into the complete graph
. In this paper, we prove this conjecture for large .Comment: 37 pages, 4 figure
Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours.
In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian
path. Although this conjecture turned out to be false, it was widely believed that such a colouring always
contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm
this by proving that every properly edge-coloured Kn has a rainbow cycle of length n − O(n
3/4
). One of
the main ingredients of our proof, which is of independent interest, shows that the subgraph of a properly
edge-coloured Kn formed by the edges a random set of colours has a similar edge distribution as a truly
random graph with the same edge density. In particular it has very good expansion properties