136,512 research outputs found
New Computational Upper Bounds for Ramsey Numbers R(3,k)
Using computational techniques we derive six new upper bounds on the
classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <=
68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are
improvements by one over the previously best known bounds.
Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on
n vertices without independent sets of order k. The new upper bounds on R(3,k)
are obtained by completing the computation of the exact values of e(3,k,n) for
all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower
bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The
enumeration of all graphs witnessing the values of e(3,k,n) is completed for
all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35
vertices is unique up to isomorphism. For the case of R(3,10), first we
establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently,
that if R(3,10) = 43 then every critical graph is regular of degree 9. Then,
using computations, we disprove the existence of the latter, and thus show that
R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for
R(3,11); added some note
Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and
how excluding one fixed cycle length affects the number of colours
required as . For vertex-colouring and , if any two
distinct vertices connected by a path of at most edges are required to be
coloured differently, then a reduction by a logarithmic (in ) factor against
the trivial bound can be obtained by excluding an odd cycle length
if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by
a path of fewer than edges are required to be coloured differently, then
excluding an even cycle length is sufficient for a logarithmic
factor reduction. For , neither of the above statements are possible
for other parity combinations of and . These results can be
considered extensions of results due to Johansson (1996) and Mahdian (2000),
and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang
(2014).Comment: 14 pages, 1 figur
Edge-coloring via fixable subgraphs
Many graph coloring proofs proceed by showing that a minimal counterexample
to the theorem being proved cannot contain certain configurations, and then
showing that each graph under consideration contains at least one such
configuration; these configurations are called \emph{reducible} for that
theorem. (A \emph{configuration} is a subgraph , along with specified
degrees in the original graph for each vertex of .)
We give a general framework for showing that configurations are reducible for
edge-coloring. A particular form of reducibility, called \emph{fixability}, can
be considered without reference to a containing graph. This has two key
benefits: (i) we can now formulate necessary conditions for fixability, and
(ii) the problem of fixability is easy for a computer to solve. The necessary
condition of \emph{superabundance} is sufficient for multistars and we
conjecture that it is sufficient for trees as well, which would generalize the
powerful technique of Tashkinov trees.
Via computer, we can generate thousands of reducible configurations, but we
have short proofs for only a small fraction of these. The computer can write
\LaTeX\ code for its proofs, but they are only marginally enlightening and can
run thousands of pages long. We give examples of how to use some of these
reducible configurations to prove conjectures on edge-coloring for small
maximum degree. Our aims in writing this paper are (i) to provide a common
context for a variety of reducible configurations for edge-coloring and (ii) to
spur development of methods for humans to understand what the computer already
knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure
Compact Routing on Internet-Like Graphs
The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing
scheme delivering a nearly optimal local memory upper bound. Using both direct
analysis and simulation, we calculate the stretch distribution of this routing
scheme on random graphs with power-law node degree distributions, . We find that the average stretch is very low and virtually
independent of . In particular, for the Internet interdomain graph,
, the average stretch is around 1.1, with up to 70% of paths
being shortest. As the network grows, the average stretch slowly decreases. The
routing table is very small, too. It is well below its upper bounds, and its
size is around 50 records for -node networks. Furthermore, we find that
both the average shortest path length (i.e. distance) and width of
the distance distribution observed in the real Internet inter-AS graph
have values that are very close to the minimums of the average stretch in the
- and -directions. This leads us to the discovery of a unique
critical quasi-stationary point of the average TZ stretch as a function of
and . The Internet distance distribution is located in a
close neighborhood of this point. This observation suggests the analytical
structure of the average stretch function may be an indirect indicator of some
hidden optimization criteria influencing the Internet's interdomain topology
evolution.Comment: 29 pages, 16 figure
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