5 research outputs found
Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments
A long-standing conjecture of Kelly states that every regular tournament on n
vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove
this conjecture for large n. In fact, we prove a far more general result, based
on our recent concept of robust expansion and a new method for decomposing
graphs. We show that every sufficiently large regular digraph G on n vertices
whose degree is linear in n and which is a robust outexpander has a
decomposition into edge-disjoint Hamilton cycles. This enables us to obtain
numerous further results, e.g. as a special case we confirm a conjecture of
Erdos on packing Hamilton cycles in random tournaments. As corollaries to the
main result, we also obtain several results on packing Hamilton cycles in
undirected graphs, giving e.g. the best known result on a conjecture of
Nash-Williams. We also apply our result to solve a problem on the domination
ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by
Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust
decomposition lemma' for application in subsequent paper
Hamilton decompositions of regular tournaments
We show that every sufficiently large regular tournament can almost
completely be decomposed into edge-disjoint Hamilton cycles. More precisely,
for each \eta>0 every regular tournament G of sufficiently large order n
contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an
approximate solution to a conjecture of Kelly from 1968. Our result also
extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our
main result. To appear in the Proceedings of the LM
Hamilton decompositions of regular bipartite tournaments
A regular bipartite tournament is an orientation of a complete balanced
bipartite graph where every vertex has its in- and outdegree both
equal to . In 1981, Jackson conjectured that any regular bipartite
tournament can be decomposed into Hamilton cycles. We prove this conjecture for
all sufficiently large bipartite tournaments. Along the way, we also prove
several further results, including a conjecture of Liebenau and Pehova on
Hamilton decompositions of dense bipartite digraphs.Comment: 119 pages, 4 figure
Path and cycle decompositions of graphs and digraphs
In this thesis, we make progress on five long standing conjectures on path and cycle decompositions of graphs and digraphs. Firstly, we confirm a conjecture of Jackson from 1981 by showing that the edges of any sufficiently large regular bipartite tournament can be decomposed into Hamilton cycles. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.
Secondly, we determine the minimum number of paths required to decompose the edges of any sufficiently large tournament of even order, thus resolving a conjecture of Alspach, Mason, and Pullman from 1976. We also prove an asymptotically optimal result for tournaments of odd order.
Finally, we give asymptotically best possible upper bounds on the minimum number of paths, cycles, and cycles and edges required to decompose the edges of any sufficiently large dense graph. This makes progress on three famous conjectures from the 1960s: Gallai's conjecture, Hajós' conjecture, and the Erdős-Gallai conjecture, respectively.
This includes joint work with António Girão, Daniela Kühn, Allan Lo, and Deryk Osthus
Matrix Graph Grammars
This book objective is to develop an algebraization of graph grammars.
Equivalently, we study graph dynamics. From the point of view of a computer
scientist, graph grammars are a natural generalization of Chomsky grammars for
which a purely algebraic approach does not exist up to now. A Chomsky (or
string) grammar is, roughly speaking, a precise description of a formal
language (which in essence is a set of strings). On a more discrete
mathematical style, it can be said that graph grammars -- Matrix Graph Grammars
in particular -- study dynamics of graphs. Ideally, this algebraization would
enforce our understanding of grammars in general, providing new analysis
techniques and generalizations of concepts, problems and results known so far.Comment: 321 pages, 75 figures. This book has is publisehd by VDM verlag, ISBN
978-363921255