9 research outputs found
Hamilton cycles in sparse robustly expanding digraphs
The notion of robust expansion has played a central role in the solution of
several conjectures involving the packing of Hamilton cycles in graphs and
directed graphs. These and other results usually rely on the fact that every
robustly expanding (di)graph with suitably large minimum degree contains a
Hamilton cycle. Previous proofs of this require Szemer\'edi's Regularity Lemma
and so this fact can only be applied to dense, sufficiently large robust
expanders. We give a proof that does not use the Regularity Lemma and, indeed,
we can apply our result to suitable sparse robustly expanding digraphs.Comment: Accepted for publication in The Electronic Journal of Combinatoric
Hamilton cycles in quasirandom hypergraphs
We show that, for a natural notion of quasirandomness in -uniform
hypergraphs, any quasirandom -uniform hypergraph on vertices with
constant edge density and minimum vertex degree contains a
loose Hamilton cycle. We also give a construction to show that a -uniform
hypergraph satisfying these conditions need not contain a Hamilton -cycle
if divides . The remaining values of form an interesting
open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm
A Dirac-type theorem for arbitrary Hamiltonian -linked digraphs
Given any digraph , let be the family of all directed
paths in , and let be a digraph with the arc set . The digraph is called arbitrary Hamiltonian -linked if for any
injective mapping and any integer set
with for each , there exists a mapping such that for
every arc , is a directed path from to of length
, and different arcs are mapped into internally vertex-disjoint directed
paths in , and . In this paper, we prove
that for any digraph with arcs and , every digraph of
sufficiently large order with minimum in- and out-degree at least
is arbitrary Hamiltonian -linked. Furthermore, we show that the lower bound
is best possible. Our main result extends some work of K\"{u}hn and Osthus et
al. \cite{20081,20082} and Ferrara, Jacobson and Pfender \cite{Jacobson}.
Besides, as a corollary of our main theorem, we solve a conjecture of Wang
\cite{Wang} for sufficiently large graphs
Cycle partitions of regular graphs
Magnant and Martin conjectured that the vertex set of any -regular graph
on vertices can be partitioned into paths (there exists a
simple construction showing that this bound would be best possible). We prove
this conjecture when , improving a result of Han, who showed
that in this range almost all vertices of can be covered by
vertex-disjoint paths. In fact, our proof gives a partition of into
cycles. We also show that, if and is bipartite, then
can be partitioned into paths (this bound in tight for bipartite
graphs).Comment: 31 pages, 1 figur
Decomposing tournaments into paths
We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kühn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament T . There is a natural lower bound for this number in terms of the degree sequence of T and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them