16,723 research outputs found
Successive vertex orderings of fully regular graphs
A graph G = (V,E) is called fully regular if for every independent set
, the number of vertices in I that are not connected
to any element of I depends only on the size of I. A linear ordering of the
vertices of G is called successive if for every i, the first i vertices induce
a connected subgraph of G. We give an explicit formula for the number of
successive vertex orderings of a fully regular graph.
As an application of our results, we give alternative proofs of two theorems
of Stanley and Gao + Peng, determining the number of linear edge orderings of
complete graphs and complete bipartite graphs, respectively, with the property
that the first i edges induce a connected subgraph. As another application, we
give a simple product formula for the number of linear orderings of the
hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i,
the first i hyperedges induce a connected subgraph. We found similar formulas
for complete (non-partite) 3-uniform hypergraphs and in another closely related
case, but we managed to verify them only when the number of vertices is small.Comment: 14 page
EDGE-ORDERED RAMSEY NUMBERS
We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges.
The edge-ordered Ramsey number R_e(G) of an edge-ordered graph G is the minimum positive integer N such that there exists an edge-ordered complete graph K_N on N vertices such that every 2-coloring of the edges of K_N contains a monochromatic copy of G as an edge-ordered subgraph of K_N.
We prove that the edge-ordered Ramsey number R_e(G) is finite for every edge-ordered graph G and we obtain better estimates for special classes of edge-ordered graphs.
In particular, we prove R_e(G) <= 2^{O(n^3\log{n})} for every bipartite edge-ordered graph G on n vertices.
We also introduce a natural class of edge-orderings, called \emph{lexicographic edge-orderings}, for which we can prove much better upper bounds on the corresponding edge-ordered Ramsey numbers
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
Consistent random vertex-orderings of graphs
Given a hereditary graph property , consider distributions of
random orderings of vertices of graphs that are preserved
under isomorphisms and under taking induced subgraphs. We show that for many
properties the only such random orderings are uniform, and give
some examples of non-uniform orderings when they exist
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
Bounded Representations of Interval and Proper Interval Graphs
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of
recognition called the bounded representation problem which we study for the
classes of interval and proper interval graphs. The input gives a graph G and
in addition for each vertex v two intervals L_v and R_v called bounds. We ask
whether there exists a bounded representation in which each interval I_v has
its left endpoint in L_v and its right endpoint in R_v. We show that the
problem can be solved in linear time for interval graphs and in quadratic time
for proper interval graphs.
Robert's Theorem states that the classes of proper interval graphs and unit
interval graphs are equal. Surprisingly the bounded representation problem is
polynomially solvable for proper interval graphs and NP-complete for unit
interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the
proper and unit interval representations behave very differently.
The bounded representation problem belongs to a wider class of restricted
representation problems. These problems are generalizations of the
well-understood recognition problem, and they ask whether there exists a
representation of G satisfying some additional constraints. The bounded
representation problems generalize many of these problems
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