7 research outputs found
Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs
P\'osa's theorem states that any graph whose degree sequence satisfies for all has a Hamilton cycle.
This degree condition is best possible. We show that a similar result holds for
suitable subgraphs of random graphs, i.e. we prove a `resilience version'
of P\'osa's theorem: if and the -th vertex degree (ordered
increasingly) of is at least for all ,
then has a Hamilton cycle. This is essentially best possible and
strengthens a resilience version of Dirac's theorem obtained by Lee and
Sudakov.
Chv\'atal's theorem generalises P\'osa's theorem and characterises all degree
sequences which ensure the existence of a Hamilton cycle. We show that a
natural guess for a resilience version of Chv\'atal's theorem fails to be true.
We formulate a conjecture which would repair this guess, and show that the
corresponding degree conditions ensure the existence of a perfect matching in
any subgraph of which satisfies these conditions. This provides an
asymptotic characterisation of all degree sequences which resiliently guarantee
the existence of a perfect matching.Comment: To appear in the Electronic Journal of Combinatorics. This version
corrects a couple of typo
Dirac's theorem for random regular graphs
We prove a `resilience' version of Dirac's theorem in the setting of random
regular graphs. More precisely, we show that, whenever is sufficiently
large compared to , a.a.s. the following holds: let be any
subgraph of the random -vertex -regular graph with minimum
degree at least . Then is Hamiltonian.
This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result
is best possible: firstly, the condition that is large cannot be omitted,
and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability &
Computin
Packing, counting and covering Hamilton cycles in random directed graphs
A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so-called Posá ‘rotation-extension’ technique for the undirected analogue. Let D(n, p) denote the random digraph on vertex set [n], obtained by adding each directed edge independently with probability p. Here we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability p > logC(n)/n. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case