42,087 research outputs found
Acyclic edge-coloring using entropy compression
An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G
and every cycle contains at least three colors. We prove that every graph with
maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4
colors, improving the previous bound of 9.62 (Delta - 1). Our bound results
from the analysis of a very simple randomised procedure using the so-called
entropy compression method. We show that the expected running time of the
procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices
and edges of G. Such a randomised procedure running in expected polynomial time
was only known to exist in the case where at least 16 Delta colors were
available. Our aim here is to make a pedagogic tutorial on how to use these
ideas to analyse a broad range of graph coloring problems. As an application,
also show that every graph with maximum degree Delta has a star coloring with 2
sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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