639 research outputs found
Algebraic Characterization of Uniquely Vertex Colorable Graphs
The study of graph vertex colorability from an algebraic perspective has
introduced novel techniques and algorithms into the field. For instance, it is
known that -colorability of a graph is equivalent to the condition for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this
paper, we extend this result by proving a general decomposition theorem for
. This theorem allows us to give an algebraic characterization of
uniquely -colorable graphs. Our results also give algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
Exhaustive generation of -critical -free graphs
We describe an algorithm for generating all -critical -free
graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove
that there are only finitely many -critical -free graphs, for
both and . We also show that there are only finitely many
-critical graphs -free graphs. For each case of these cases we
also give the complete lists of critical graphs and vertex-critical graphs.
These results generalize previous work by Hell and Huang, and yield certifying
algorithms for the -colorability problem in the respective classes.
Moreover, we prove that for every , the class of 4-critical planar
-free graphs is finite. We also determine all 27 4-critical planar
-free graphs.
We also prove that every -free graph of girth at least five is
3-colorable, and determine the smallest 4-chromatic -free graph of
girth five. Moreover, we show that every -free graph of girth at least
six and every -free graph of girth at least seven is 3-colorable. This
strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with
arXiv:1504.0697
The Complexity of the List Partition Problem for Graphs
The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (i≠j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete
On the critical exponents of random k-SAT
There has been much recent interest in the satisfiability of random Boolean
formulas. A random k-SAT formula is the conjunction of m random clauses, each
of which is the disjunction of k literals (a variable or its negation). It is
known that when the number of variables n is large, there is a sharp transition
from satisfiability to unsatisfiability; in the case of 2-SAT this happens when
m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The
sharpness of this transition is characterized by a critical exponent, sometimes
called \nu=\nu_k (the smaller the value of \nu the sharper the transition).
Experiments have suggested that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05,
\nu_5=1.1+-0.05, \nu_6 = 1.05+-0.05, and heuristics have suggested that \nu_k
--> 1 as k --> infinity. We give here a simple proof that each of these
exponents is at least 2 (provided the exponent is well-defined). This result
holds for each of the three standard ensembles of random k-SAT formulas: m
clauses selected uniformly at random without replacement, m clauses selected
uniformly at random with replacement, and each clause selected with probability
p independent of the other clauses. We also obtain similar results for
q-colorability and the appearance of a q-core in a random graph.Comment: 11 pages. v2 has revised introduction and updated reference
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