11 research outputs found
Nowhere-zero flows and structures in cubic graphs
Wir widerlegen zwei Vermutungen, die im Zusammenhang mit Kreisüberdeckungen von kubischen Graphen stehen. Die erste Vermutung, welche kubische Graphen mit dominierenden Kreisen betrifft, widerlegen wir durch Erweiterung eines Theorems von Gallai über induzierte eulersche Graphen und durch Konstruktion spezieller snarks. Die zweite Vermutung, welche frames betrifft, widerlegen wir durch Betrachtung der Frage nach der Existenz von speziellen spannenden Teilgraphen in 3-fach zusammenhängenden kubischen Graphen. Weiters übersetzen wir Probleme über Flüsse in kubischen Graphen in Knotenfärbungsprobleme von planaren Graphen und erhalten eine neue Charakterisierung von snarks. Schliesslich verbessern und erweitern wir Resultate über Knotenfärbungsprobleme in Quadrangulierungen. Zu Ende stellen wir neue Vermutungen auf, die im Zusammenhang mit Kreisüberdeckungen und Strukturen in kubischen Graphen stehen.We disprove two conjectures which are related to cycle double cover problems. The first conjecture concerns cubic graphs with dominating cycle. We disprove this conjecture by extending a result of Gallai about induced eulerian subgraphs and by constructing special snarks. The second conjecture concerns frames. We show that this conjecture is false by considering the problem whether every 3-connected cubic graph has a spanning subgraph with certain properties. Moreover, we transform flow-problems of cubic graphs into vertex coloring problems of plane graphs. We obtain thereby a new characterization of snarks. Furthermore, we improve and extend results about vertex coloring problems of quadrangulations. Finally we pose new problems and state conjectures which are related to cycle double covers and structures in cubic graphs
Sandwiching saturation number of fullerene graphs
The saturation number of a graph is the cardinality of any smallest
maximal matching of , and it is denoted by . Fullerene graphs are
cubic planar graphs with exactly twelve 5-faces; all the other faces are
hexagons. They are used to capture the structure of carbon molecules. Here we
show that the saturation number of fullerenes on vertices is essentially
Numerical Linear Algebra applications in Archaeology: the seriation and the photometric stereo problems
The aim of this thesis is to explore the application of Numerical Linear Algebra to Archaeology. An ordering problem called the seriation problem, used for dating findings and/or artifacts deposits, is analysed in terms of graph theory. In particular, a Matlab implementation of an algorithm for spectral seriation, based on the use of the Fiedler vector of the Laplacian matrix associated with the problem, is presented. We consider bipartite graphs for describing the seriation problem, since the interrelationship between the units (i.e. archaeological sites) to be reordered, can be described in terms of these graphs. In our archaeological metaphor of seriation, the two disjoint nodes sets into which the vertices of a bipartite graph can be divided, represent the excavation sites and the artifacts found inside
them.
Since it is a difficult task to determine the closest bipartite network to a given one, we describe how a starting network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems.
Another numerical problem related to Archaeology is the 3D reconstruction of the shape of an object from a set of digital pictures. In particular, the Photometric Stereo (PS) photographic technique is considered
Round and Bipartize for Vertex Cover Approximation
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (?, S), consisting of a graph with an odd cycle transversal.
If S is a stable set, we prove a tight approximation ratio of 1 + 1/?, where 2? -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph ?? : = ?/S and satisfies ? ? [2,?], with ? = ? corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/?) (1 - ?) + 2 ?, where ? ? [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph ??, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ? and ?, which are ? = 2 and ? = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph
Bipartizing fullerenes
A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such
that all other faces are 6-faces. We show that any fullerene graph on n
vertices can be bipartized by removing O(sqrt{n}) edges. This bound is
asymptotically optimal.Comment: 14 pages, 4 figure
Research Problems from the BCC21
AbstractA collection of open problems, mostly presented at the problem session of the 21st British Combinatorial Conference
Round and bipartize for vertex cover approximation
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (G, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/ρ, where 2ρ − 1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph G̃:= G/S and satisfies ρ ∈ [2, ∞], with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1 + 1/ρ) (1 − α) + 2α, where α ∈ [0, 1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph G̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 − 4/n, recover the integrality gap of 2 − 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph
Realizations of multiassociahedra via bipartite rigidity
Let denote the simplicial complex of -crossing-free subsets
of edges in . Here and . It is
conjectured that this simplicial complex is polytopal (Jonsson 2005). However,
despite several recent advances, this is still an open problem.
In this paper we attack this problem using as a vector configuration the rows
of a rigidity matrix, namely, hyperconnectivity restricted to bipartite graphs.
We see that in this way can be realized as a polytope for and
, and as a fan for and , and for and .
However, we also prove that the cases with and
are not realizable in this way.
We also give an algebraic interpretation of the rigidity matroid, relating it
to a projection of determinantal varieties with implications in matrix
completion, and prove the presence of a fan isomorphic to in
the tropicalization of that variety.Comment: 30 pages, 2 figures. arXiv admin note: text overlap with
arXiv:2212.1426