8,740 research outputs found
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
Graph coloring with no large monochromatic components
For a graph G and an integer t we let mcc_t(G) be the smallest m such that
there exists a coloring of the vertices of G by t colors with no monochromatic
connected subgraph having more than m vertices. Let F be any nontrivial
minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any
n-vertex graph G \in F. This bound is asymptotically optimal and it is attained
for planar graphs. More generally, for every such F and every fixed t we show
that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G
with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}).
It is also interesting to consider graphs of bounded degrees. Haxell, Szabo,
and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5.
We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n),
and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex
graphs of maximum degree 7, average degree at most 6+\epsilon for all
subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only
that the maximum order of magnitude of \mcc_2 is between \sqrt n and n.
We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure
Two novel evolutionary formulations of the graph coloring problem
We introduce two novel evolutionary formulations of the problem of coloring
the nodes of a graph. The first formulation is based on the relationship that
exists between a graph's chromatic number and its acyclic orientations. It
views such orientations as individuals and evolves them with the aid of
evolutionary operators that are very heavily based on the structure of the
graph and its acyclic orientations. The second formulation, unlike the first
one, does not tackle one graph at a time, but rather aims at evolving a
`program' to color all graphs belonging to a class whose members all have the
same number of nodes and other common attributes. The heuristics that result
from these formulations have been tested on some of the Second DIMACS
Implementation Challenge benchmark graphs, and have been found to be
competitive when compared to the several other heuristics that have also been
tested on those graphs.Comment: To appear in Journal of Combinatorial Optimizatio
Extremal Optimization at the Phase Transition of the 3-Coloring Problem
We investigate the phase transition of the 3-coloring problem on random
graphs, using the extremal optimization heuristic. 3-coloring is among the
hardest combinatorial optimization problems and is closely related to a 3-state
anti-ferromagnetic Potts model. Like many other such optimization problems, it
has been shown to exhibit a phase transition in its ground state behavior under
variation of a system parameter: the graph's mean vertex degree. This phase
transition is often associated with the instances of highest complexity. We use
extremal optimization to measure the ground state cost and the ``backbone'', an
order parameter related to ground state overlap, averaged over a large number
of instances near the transition for random graphs of size up to 512. For
graphs up to this size, benchmarks show that extremal optimization reaches
ground states and explores a sufficient number of them to give the correct
backbone value after about update steps. Finite size scaling gives
a critical mean degree value . Furthermore, the
exploration of the degenerate ground states indicates that the backbone order
parameter, measuring the constrainedness of the problem, exhibits a first-order
phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available
at http://www.physics.emory.edu/faculty/boettcher
Random Graphs with Hidden Color
We propose and investigate a unifying class of sparse random graph models,
based on a hidden coloring of edge-vertex incidences, extending an existing
approach, Random graphs with a given degree distribution, in a way that admits
a nontrivial correlation structure in the resulting graphs.
The approach unifies a number of existing random graph ensembles within a
common general formalism, and allows for the analytic calculation of observable
graph characteristics.
In particular, generating function techniques are used to derive the size
distribution of connected components (clusters) as well as the location of the
percolation threshold where a giant component appears.Comment: 4 pages, no figures, RevTe
The min-max edge q-coloring problem
In this paper we introduce and study a new problem named \emph{min-max edge
-coloring} which is motivated by applications in wireless mesh networks. The
input of the problem consists of an undirected graph and an integer . The
goal is to color the edges of the graph with as many colors as possible such
that: (a) any vertex is incident to at most different colors, and (b) the
maximum size of a color group (i.e. set of edges identically colored) is
minimized. We show the following results: 1. Min-max edge -coloring is
NP-hard, for any . 2. A polynomial time exact algorithm for min-max
edge -coloring on trees. 3. Exact formulas of the optimal solution for
cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial
lower bound of the optimal solution with respect to the average degree of the
graph. 5. An approximation algorithm for planar graphs.Comment: 16 pages, 5 figure
Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
We consider the problem of coloring Erdos-Renyi and regular random graphs of
finite connectivity using q colors. It has been studied so far using the cavity
approach within the so-called one-step replica symmetry breaking (1RSB) ansatz.
We derive a general criterion for the validity of this ansatz and, applying it
to the ground state, we provide evidence that the 1RSB solution gives exact
threshold values c_q for the q-COL/UNCOL phase transition. We also study the
asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in
perfect agreement with rigorous mathematical bounds, as well as the nature of
excited states, and give a global phase diagram of the problem.Comment: 23 pages, 10 figures. Replaced with accepted versio
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