2,501 research outputs found
A semi-exact degree condition for Hamilton cycles in digraphs
The paper is concerned with directed versions of Posa's theorem and Chvatal's
theorem on Hamilton cycles in graphs.
We show that for each a>0, every digraph G of sufficiently large order n
whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq
>... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In
fact, we can weaken these assumptions to
(i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^-
\geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G
is Hamiltonian. This provides an approximate version of a conjecture of
Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and
Treglown
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric
graphs: after suitably metrising the space of graph partitions, we restrict
Laplacians to the clusters thus arising and use their spectral gaps to define
several notions of partition energies; this is the graph counterpart of the
well-known theory of spectral minimal partitions on planar domains and includes
the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012),
815--838] as a special case. We focus on the existence of optimisers for a
large class of functionals defined on such partitions, but also study their
qualitative properties, including stability, regularity, and parameter
dependence. We also discuss in detail their interplay with the theory of nodal
partitions. Unlike in the case of domains, the one-dimensional setting of
metric graphs allows for explicit computation and analytic -- rather than
numerical -- results. Not only do we recover the main assertions in the theory
of spectral minimal partitions on domains, as studied in [Conti \textit{et al},
Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\
Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we
can also generalise some of them and answer (the graph counterparts of) a few
open questions
Resolution of the Oberwolfach problem
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of
into edge-disjoint copies of a given -factor. We show that this
can be achieved for all large . We actually prove a significantly more
general result, which allows for decompositions into more general types of
factors. In particular, this also resolves the Hamilton-Waterloo problem for
large .Comment: 28 page
Partitioning 3-colored complete graphs into three monochromatic cycles
We show in this paper that in every 3-coloring of the edges of Kn all but o(n)
of its vertices can be partitioned into three monochromatic cycles. From this, using
our earlier results, actually it follows that we can partition all the vertices into at
most 17 monochromatic cycles, improving the best known bounds. If the colors of
the three monochromatic cycles must be different then one can cover ( 3
4 â o(1))n
vertices and this is close to best possible
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