561 research outputs found
Improved bounds on coloring of graphs
Given a graph with maximum degree , we prove that the
acyclic edge chromatic number of is such that . Moreover we prove that:
if has girth ; a'(G)\le
\lceil5.77 (\Delta-1)\rc if
has girth ; a'(G)\le \lc4.52(\D-1)\rc if ;
a'(G)\le \D+2\, if g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil.
We further prove that the acyclic (vertex) chromatic number of is
such that
a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the
star-chromatic number of is such that \chi_s(G)\le
\lc4.34\Delta^{3/2}+ 1.5\D\rc. We finally prove that the \b-frugal chromatic
number \chi^\b(G) of is such that \chi^\b(G)\le \lc\max\{k_1(\b)\D,\;
k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc, where k_1(\b) and k_2(\b) are
decreasing functions of \b such that k_1(\b)\in[4, 6] and
k_2(\b)\in[2,5].
To obtain these results we use an improved version of the Lov\'asz Local
Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.Comment: Introduction revised. Added references. Corrected typos. Proof of
Theorem 2 (items c-f) written in more detail
Linear transformation distance for bichromatic matchings
Let be a set of points in general position, where is a
set of blue points and a set of red points. A \emph{-matching}
is a plane geometric perfect matching on such that each edge has one red
endpoint and one blue endpoint. Two -matchings are compatible if their
union is also plane.
The \emph{transformation graph of -matchings} contains one node for each
-matching and an edge joining two such nodes if and only if the
corresponding two -matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of for its diameter, which is asymptotically
tight
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
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