4,955 research outputs found
Edge-decompositions of graphs with high minimum degree
A fundamental theorem of Wilson states that, for every graph , every
sufficiently large -divisible clique has an -decomposition. Here a graph
is -divisible if divides and the greatest common divisor
of the degrees of divides the greatest common divisor of the degrees of
, and has an -decomposition if the edges of can be covered by
edge-disjoint copies of . We extend this result to graphs which are
allowed to be far from complete. In particular, together with a result of
Dross, our results imply that every sufficiently large -divisible graph of
minimum degree at least has a -decomposition. This
significantly improves previous results towards the long-standing conjecture of
Nash-Williams that every sufficiently large -divisible graph with minimum
degree at least has a -decomposition. We also obtain the
asymptotically correct minimum degree thresholds of for the
existence of a -decomposition, and of for the existence of a
-decomposition, where . Our main contribution is a
general `iterative absorption' method which turns an approximate or fractional
decomposition into an exact one. In particular, our results imply that in order
to prove an asymptotic version of Nash-Williams' conjecture, it suffices to
show that every -divisible graph with minimum degree at least
has an approximate -decomposition,Comment: 41 pages. This version includes some minor corrections, updates and
improvement
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs
Drawings of highly connected (dense) graphs can be very difficult to read.
Power Graph Analysis offers an alternate way to draw a graph in which sets of
nodes with common neighbours are shown grouped into modules. An edge connected
to the module then implies a connection to each member of the module. Thus, the
entire graph may be represented with much less clutter and without loss of
detail. A recent experimental study has shown that such lossless compression of
dense graphs makes it easier to follow paths. However, computing optimal power
graphs is difficult. In this paper, we show that computing the optimal
power-graph with only one module is NP-hard and therefore likely NP-hard in the
general case. We give an ILP model for power graph computation and discuss why
ILP and CP techniques are poorly suited to the problem. Instead, we are able to
find optimal solutions much more quickly using a custom search method. We also
show how to restrict this type of search to allow only limited back-tracking to
provide a heuristic that has better speed and better results than previously
known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of
the same nam
On covering expander graphs by Hamilton cycles
The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree satisfies some basic expansion properties and contains
a family of edge disjoint Hamilton cycles, then there also
exists a covering of its edges by Hamilton cycles. This
implies that for every and every there exists
a covering of all edges of by Hamilton cycles
asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275
- …