2,240 research outputs found
The Cycle Spectrum of Claw-free Hamiltonian Graphs
If is a claw-free hamiltonian graph of order and maximum degree
with , then has cycles of at least many different lengths.Comment: 9 page
Long paths and cycles in random subgraphs of graphs with large minimum degree
For a given finite graph of minimum degree at least , let be a
random subgraph of obtained by taking each edge independently with
probability . We prove that (i) if for a function
that tends to infinity as does, then
asymptotically almost surely contains a cycle (and thus a path) of length at
least , and (ii) if , then
asymptotically almost surely contains a path of length at least . Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking to be the complete graph on vertices.Comment: 26 page
Precise Partitions Of Large Graphs
First by using an easy application of the Regularity Lemma, we extend some known results about cycles of many lengths to include a specified edge on the cycles. The results in this chapter will help us in rest of this thesis. In 2000, Enomoto and Ota posed a conjecture on the existence of path decomposition of graphs with fixed start vertices and fixed lengths. We prove this conjecture when |G| is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices. Furthermore, sharp minimum degree and degree sum conditions are proven for the existance of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs. Finally, we prove a sharp connectivity and degree sum condition for the existence of a subdivision of a multigraph in which some of the vertices are specified and the distance between each pair of vertices in the subdivision is prescribed (within one)
Matchings and Hamilton Cycles with Constraints on Sets of Edges
The aim of this paper is to extend and generalise some work of Katona on the
existence of perfect matchings or Hamilton cycles in graphs subject to certain
constraints. The most general form of these constraints is that we are given a
family of sets of edges of our graph and are not allowed to use all the edges
of any member of this family. We consider two natural ways of expressing
constraints of this kind using graphs and using set systems.
For the first version we ask for conditions on regular bipartite graphs
and for there to exist a perfect matching in , no two edges of which
form a -cycle with two edges of .
In the second, we ask for conditions under which a Hamilton cycle in the
complete graph (or equivalently a cyclic permutation) exists, with the property
that it has no collection of intervals of prescribed lengths whose union is an
element of a given family of sets. For instance we prove that the smallest
family of -sets with the property that every cyclic permutation of an
-set contains two adjacent pairs of points has size between
and . We also give bounds on the general version of this problem
and on other natural special cases.
We finish by raising numerous open problems and directions for further study.Comment: 21 page
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
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