77,841 research outputs found
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
General -position sets
The general -position number of a graph is the
cardinality of a largest set for which no three distinct vertices from
lie on a common geodesic of length at most . This new graph parameter
generalizes the well studied general position number. We first give some
results concerning the monotonic behavior of with respect to
the suitable values of . We show that the decision problem concerning
finding is NP-complete for any value of . The value of when is a path or a cycle is computed and a structural
characterization of general -position sets is shown. Moreover, we present
some relationships with other topics including strong resolving graphs and
dissociation sets. We finish our exposition by proving that is
infinite whenever is an infinite graph and is a finite integer.Comment: 16 page
On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof
We exhibit a family of graphs that develop turning singularities (i.e. their
Lipschitz seminorm blows up and they cease to be a graph, passing from the
stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem
where the permeability is given by a nonnegative step function. We study the
influence of different choices of the permeability and different boundary
conditions (both at infinity and considering finite/infinite depth) in the
development or prevention of singularities for short time. In the general case
(inhomogeneous, confined) we prove a bifurcation diagram concerning the
appearance or not of singularities when the depth of the medium and the
permeabilities change. The proofs are carried out using a combination of
classical analysis techniques and computer-assisted verification.Comment: 30 pages, 6 figure
Extremal graph colouring and tiling problems
In this thesis, we study a variety of different extremal graph colouring and tiling problems in finite and infinite graphs.
Confirming a conjecture of Gyárfás, we show that for all k, r ∈ N there is a constant C > 0 such that the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a collection of at most C monochromatic tight cycles. We shall say that the family of tight cycles has finite r-colour tiling number. We further prove that, for all natural numbers k, p and r, the family of p-th powers of k-uniform tight cycles has finite r-colour tiling number. The case where k = 2 settles a problem of Elekes, Soukup, Soukup and Szentmiklóssy. We then show that for all natural numbers ∆, r, every family F = {F1, F2, . . .} of graphs with v (Fn) = n and ∆(Fn) ≤ ∆ for every n ∈ N has finite r-colour tiling number. This makes progress on a conjecture of Grinshpun and Sárközy.
We study Ramsey problems for infinite graphs and prove that in every 2-edge- colouring of KN, the countably infinite complete graph, there exists a monochromatic infinite path P such that V (P) has upper density at least (12 + √8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdős and Galvin. We study similar problems for many other graphs including trees and graphs of bounded degree or degeneracy and prove analogues of many results concerning graphs with linear Ramsey number in finite Ramsey theory.
We also study a different sort of tiling problem which combines classical problems from extremal and probabilistic graph theory, the Corrádi–Hajnal theorem and (a special case of) the Johansson–Kahn–Vu theorem. We prove that there is some constant C > 0 such that the following is true for every n ∈ 3N and every p ≥ Cn−2/3 (log n)1/3. If G is a graph on n vertices with minimum degree at least 2n/3, then Gp (the random subgraph of G obtained by keeping every edge independently with probability p) contains a triangle tiling with high probability
Random walks on graphs: ideas, techniques and results
Random walks on graphs are widely used in all sciences to describe a great
variety of phenomena where dynamical random processes are affected by topology.
In recent years, relevant mathematical results have been obtained in this
field, and new ideas have been introduced, which can be fruitfully extended to
different areas and disciplines. Here we aim at giving a brief but
comprehensive perspective of these progresses, with a particular emphasis on
physical aspects.Comment: LateX file, 34 pages, 13 jpeg figures, Topical Revie
Probabilistic regular graphs
Deterministic graph grammars generate regular graphs, that form a structural
extension of configuration graphs of pushdown systems. In this paper, we study
a probabilistic extension of regular graphs obtained by labelling the terminal
arcs of the graph grammars by probabilities. Stochastic properties of these
graphs are expressed using PCTL, a probabilistic extension of computation tree
logic. We present here an algorithm to perform approximate verification of PCTL
formulae. Moreover, we prove that the exact model-checking problem for PCTL on
probabilistic regular graphs is undecidable, unless restricting to qualitative
properties. Our results generalise those of EKM06, on probabilistic pushdown
automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611
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