6,597 research outputs found
On random k-out sub-graphs of large graphs
We consider random sub-graphs of a fixed graph with large minimum
degree. We fix a positive integer and let be the random sub-graph
where each independently chooses random neighbors, making
edges in all. When the minimum degree then is -connected w.h.p. for ;
Hamiltonian for sufficiently large. When , then has
a cycle of length for . By w.h.p. we mean
that the probability of non-occurrence can be bounded by a function
(or ) where
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
Finding long cycles in graphs
We analyze the problem of discovering long cycles inside a graph. We propose
and test two algorithms for this task. The first one is based on recent
advances in statistical mechanics and relies on a message passing procedure.
The second follows a more standard Monte Carlo Markov Chain strategy. Special
attention is devoted to Hamiltonian cycles of (non-regular) random graphs of
minimal connectivity equal to three
Long cycles in graphs with large degree sums and neighborhood unions
We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings
Cycles containing many vertices of large degree
AbstractLet G be a 2-connected graph of order n, r a real number and Vr=v ϵ V(G)¦d(v)⩾r. It is shown that G contains a cycle missing at most max {0, n − 2r} vertices of Vr, yielding a common generalization of a result of Dirac and one of Shi Ronghua. A stronger conclusion holds if r⩾13(n+2): G contains a cycle C such that either V(C)⊇Vr or ¦ V(C)¦⩾2r. This theorem extends a result of Häggkvist and Jackson and is proved by first showing that if r⩾13(n+2), then G contains a cycle C for which ¦Vr∩V(C)¦is maximal such that N(x)⊆V(C) whenever x ϵ Vr − V(C) (∗). A result closely related to (∗) is stated and a result of Nash-Williams is extended using (∗)
Long cycles, degree sums and neighborhood unions
AbstractFor a graph G, define the parameters α(G)=max{|S| |S is an independent set of vertices of G}, σk(G)=min{∑ki=1d(vi)|{v1,…,vk} is an independent set} and NCk(G)= min{|∪ki=1 N(vi)∥{v1,…,vk} is an independent set} (k⩾2). It is shown that every 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,n+NCr+5+∈(n+r)(G)-α(G)}, where ε(i)=3(⌈13i⌉−13i). This result extends previous results in Bauer et al. (1989/90), Faßbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,2NC⌊18(n+6r+17)⌋(G)}. Analogous results are established for 2-connected graphs
- …