13,762 research outputs found
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
Topological lower bounds for the chromatic number: A hierarchy
This paper is a study of ``topological'' lower bounds for the chromatic
number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978,
in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology.
This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with
all -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . Several other proofs
have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz,
Greene, and others), all of them based on some version of the Borsuk--Ulam
theorem, but otherwise quite different. Each can be extended to yield some
lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe
that \emph{every} finite graph may be represented as a generalized Kneser
graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the
strongest one being essentially Lov\'asz' original bound in terms of a
neighborhood complex. We also present and compare various definitions of a
\emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz
and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but
the construction is simpler and functorial, mapping graphs with homomorphisms
to -spaces with -maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea
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