Non-contractible Cycles Research Articles

Multiple-Source Shortest Paths in Embedded Graphs

Let $G$ be a directed graph with $n$ vertices and nonnegative weights in its directed edges, embedded on a surface of genus $g$, and let $f$ be an arbitrary face of $G$. We describe a randomized algorithm to preprocess the graph in $O(gn \log n)$ time with high probability, so that the shortest-path distance from any vertex on the boundary of $f$ to any other vertex in $G$ can be retrieved in $O(\log n)$ time. Our result directly generalizes the $O(n\log n)$-time algorithm of Klein [Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of $f$. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in $O(g^2 n\log n)$ time with high probability. Our high-probability time bounds hold in the worst case for generic edge weights or with an additional $O(\log n)$ factor for arbitrary edge weights.

Black holes and singularity resolution in higher spin gravity

We investigate higher spin theories of gravity in three dimensions based on the gauge group SL(N,R)*SL(N,R). In these theories the usual diffeomorphism symmetry is enhanced to include higher spin gauge transformations under which traditional geometric notions of curvature and causality are no longer invariant. This implies, for example, that apparently singular geometries can be rendered smooth by a gauge transformation, much like the resolution of orbifold singularities in string theory. The classical solutions, including the recently constructed higher spin black hole, are characterized by their holonomies around the non-contractible cycles of space-time. The black hole solutions are shown to be gauge equivalent to a BTZ black hole which is charged under a set of U(1) Chern-Simons fields. Nevertheless, depending on the choice of embedding of the gravitational gauge group, the space-time geometry may be non-trivial. We study in detail the N=3 example, where this observation allows us to find a gauge where the black hole geometry takes a simple form and the thermodynamic properties can be studied.

Phases of a two-dimensional large-Ngauge theory on a torus

We consider two-dimensional large N gauge theory with D adjoint scalars on a torus, which is obtained from a D+2 dimensional pure Yang-Mills theory on T^{D+2} with D small radii. The two dimensional model has various phases characterized by the holonomy of the gauge field around non-contractible cycles of the 2-torus. We determine the phase boundaries and derive the order of the phase transitions using a method, developed in an earlier work (arxiv:0910.4526), which is nonperturbative in the 'tHooft coupling and uses a 1/D expansion. We embed our phase diagram in the more extensive phase structure of the D+2 dimensional Yang-Mills theory and match with the picture of a cascade of phase transitions found earlier in lattice calculations (arxiv:0710.0098). We also propose a dual gravity system based on a Scherk-Schwarz compactification of a D2 brane wrapped on a 3-torus and find a phase structure which is similar to the phase diagram found in the gauge theory calculation.

Dominating sets in triangulations on surfaces

A dominating set D ⊆ V ( G ) of a graph G is a set such that each vertex v ∈ V ( G ) is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any n -vertex plane triangulation has a dominating set of size at most n /3, and conjectured a bound of n /4 for n sufficiently large. King and Pelsmajer recently proved this for graphs with maximum degree at most 6. Plummer and Zha (2009) and Honjo, Kawarabayashi, and Nakamoto (2009) extended the n /3 bound to triangulations on surfaces. We prove two related results: (i) There is a constant c 1 such that any n -vertex plane triangulation with maximum degree at most 6 has a dominating set of size at most n /6 + c 1 . (ii) For any surface S , t ≥ 0, and ε > 0, there exists c 2 such that for any n -vertex triangulation on S with at most t vertices of degree other than 6, there is a dominating set of size at most n (1/6 + ε ) + c 2 . As part of the proof, we also show that any n -vertex triangulation of a non-orientable surface has a non-contractible cycle of length at most 2√ n . Albertson and Hutchinson (1986) proved that for n -vertex triangulation of an orientable surface other than a sphere has a non-contractible cycle of length √(2 n ), but no similar result was known for non-orientable surfaces.

Finding one tight cycle

A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O ( n log n ) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O (min{ g 3 , n }, n log n ) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O ( n log n ) time, a tight octagonal decomposition in O ( gn log n ) time, and a shortest contractible cycle enclosing a nonempty set of faces in O ( n log 2 n ) time.

Lessons from giant gravitons on AdS 5 × T 1,1

We implement Mikhailov's holomorphic curve construction to explore various properties of giant gravitons in type IIB string theory on $AdS_{5}\times T^{1,1}$. By coloring the D-brane worldvolume, we are able to show how, in the string theory, the giant graviton factorizes at its maximal size into two dibaryons - topologically stable D-branes wrapping non-contractible cycles in the $T^{1,1}$. This is related to the structure of the symmetry group of the emergent Klebanov-Witten gauge theory being a product - $SU(N) \times SU(N)$ instead of the canonical $SU(N)$. Finally, we complete this study with a systematic and detailed construction of the spectrum of small fluctuations about the giant graviton configuration. Curiously, we find that the fluctuation spectrum depends on the size of the giant. The similarity of the operator structures in the Klebanov-Witten and ABJM theories leads us to believe that the D4-brane giant graviton in type IIA string theory on $AdS_{4}\times \mathbb{CP}^{3}$ factorizes into two $\mathbb{CP}^{2}$ dibaryons in a qualitatively similar way.

Finding Shortest Non-Separating and Non-Contractible Cycles for Topologically Embedded Graphs

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in \(O(g^{3/2}V^{3/2}\log V+g^{5/2}V^{1/2})\) time, where V is the number of vertices in the graph and g is the genus of the surface. If \(g=o(V^{1/3})\), this represents an improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in \(O(g^{O(g)}V^{3/2})\) time, which improves previous results for fixed genus. This result can be applied for computing the face-width and the non-separating face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in \(O(V^{5/4}\log V)\) time.

On the number of limit cycles of some systems on the cylinder

In this article we give two criteria for bounding the number of non-contractible limit cycles of a family of differential systems on the cylinder. This family includes Abel equations as well as the polar expression of several types of planar polynomial systems given by the sum of three homogeneous vector fields.

Coloring Vertices and Faces of Locally Planar Graphs

If G is an embedded graph, a vertex-face r-coloring is a mapping that assigns a color from the set {1, . . . ,r} to every vertex and every face of G such that different colors are assigned whenever two elements are either adjacent or incident. Let χvf(G) denote the minimum r such that G has a vertex-face r-coloring. Ringel conjectured that if G is planar, then χvf(G)≤6. A graph G drawn on a surface S is said to be 1-embedded in S if every edge crosses at most one other edge. Borodin proved that if G is 1-embedded in the plane, then χ(G)≤6. This result implies Ringel's conjecture. Ringel also stated a Heawood style theorem for 1-embedded graphs. We prove a slight strengthening of this result. If G is 1-embedded in S, let w(G) denote the edge-width of G, i.e. the length of a shortest non-contractible cycle in G. We show that if G is 1-embedded in S and w(G) is large enough, then the list chromatic number ch(G) is at most 8.

(2+1)-Dimensional Charged Black Hole in Topologically Massive Electrodynamics

The 2+1 black hole coupled to a Maxwell field can be charged in two different ways. Besides a Coulomb field, whose potential grows logarithmically in the radial coordinate, there also exists a topological charge due to the existence of a noncontractible cycle. The topological charge does not gravitate and is somehow decoupled from the black hole. This situation changes if one turns on the Chern-Simons term for the Maxwell field. First, the flux integral at infinity becomes equal to the topological charge. Second, demanding regularity of the black hole horizon, the Coulomb charge must vanish identically. Hence, in 2+1 topologically massive electrodynamics coupled to gravity, the black hole can support holonomies only for the Maxwell field. This means that the charged black hole is constructed from the vacuum by means of spacetime identifications.

Extending precolorings of subgraphs of locally planar graphs

The width of an embedded graph is the length of a shortest noncontractible cycle. Suppose G is embedded in a surface S (either orientable or not) with large width. In this case G is said to be locally planar. Suppose P⊂ V( G) is a set of vertices such that the components of G[ P] are each 2-colorable, have bounded diameter and are suitably distant from each other. We show that any 5-coloring of G[ P] in which each component is 2-colored extends to a 5-coloring of all of G. Thus, for an arbitrary surface, the extension theorems for precolorings of subgraphs of locally planar graphs parallel the results for planar graphs. Crucial to the proof of this result is the nice cycle lemma, viz. If C is a minimal, noncontractible, and nonseparating cycle in a so-called orderly triangulation of at least moderate width, then there is a cycle C′ such that C′ lies within the fourth neighborhood of C, C′ is minimal, homotopic to C, and C′ either has even length or contains a vertex of degree 4. Such a nice cycle is useful in producing 5-colorings. We introduce the idea of optimal shortcuts in order to prove the nice cycle lemma and the idea of relative width in order to prove the main theorem. Our results generalize to extension theorems for precolorings with q≥3 colors.

Separating Cycles in Doubly Toroidal Embeddings

We show that every 4-representative graph embedding in the double torus contains a noncontractible cycle that separates the surface into two pieces. As a special case, every triangulation of the double torus in which every noncontractible cycle has length at least 4 has a noncontractible cycle that separates the surface into two pieces.

M-theory and stringy corrections to anti-de Sitter black holes and conformal field theories

We consider black holes in anti-de Sitter space AdS p+2 ( p = 2, 3, 5), which have hyperbolic, flat or spherical event horizons. The O ( α′ 3) corrections (or the leading corrections in powers of the eleven-dimensional Planck length, in the case of M-theory compactifications) to the black hole metrics are computed for the various topologies and dimensions. We investigate the consequences of the stringy or M-theory corrections for the black hole thermodynamics. In particular, we show the emergence of a stable branch of small spherical black holes. Surprisingly, for any of the considered dimension and topologies, the corrected thermodynamical quantities turn out to coincide with those calculated within a simplified approach, which uses only the unperturbed metric. We obtain the corrected Hawking-Page transition temperature for black holes with spherical horizons, and show that for p = 3 this phase transition disappears at a value of α′ considerably smaller than that estimated previously by Gao and Li. Using the AdS/CFT correspondence, we determine the S 1 × S 3 N = 4 SYM phase diagram for sufficiently large 't Hooft coupling, and show that the critical point at which the Hawking-Page transition disappears (the correspondence point of Horowitz-Polchinski), occurs at g 2 YM N ≈ 20.5. The d = 4 and d = 7 black hole phase diagrams are also determined, and connection is made with the corresponding boundary CFTs. Finally, for flat and hyperbolic horizons, we show that the leading stringy or M-theory corrections do not give rise to any phase transition. However, if the horizon is compactified to a torus T p or to a quotient of hyperbolic space, H p/Γ , the appearance of light winding modes around non-contractible cycles signal new phase transitions, which in the toroidal case have previously been discussed by Barbón et al. We comment on these phase transitions for SYM on H p/Γ and on T p , when the moduli of the torus are taken into account.

A simple construction of high representativity triangulations

We show a construction of high representativity triangulations of compact surfaces, equivalently, triangulations without short non-contractible cycles, and give a polynomial-time algorithm to carry out the construction. We improve substantially the previously known lower bound for the representativity of such a triangulation. More precisely, we present an O(max( g 2, n))-time algorithm that, for given g and n ( g > 1 and n > c' g log g) constructs an n-vertex triangulation of a genus g oriented surface without boundary such that the representativity of the triangulation is at least c√ n/ g √log g, where c, c′ are constants. We extend our result to nonorientable surfaces and surfaces with boundary. In the later case we replace the genus, g, of the surface with parameter g′ = g + d/2, where d is the number of boundary components.

Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997

Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997

Disjoint Essential Cycles

Graphs that have two disjoint noncontractible cycles in every possible embedding in surfaces are characterized. Similar characterization is given for the class of graphs whose orientable embeddings (embeddings in surfaces different from the projective plane, respectively) always have two disjoint noncontractible cycles. For graphs which admit embeddings in closed surfaces without having two disjoint noncontractible cycles, such embeddings are structurally characterized.

Separating and Nonseparating Disjoint Homotopic Cycles in Graph Embeddings

We show that if a graph G is embedded in a surface Σ with representativity ρ , then G contains at least ⌊( ρ −1)/2⌋ pairwise disjoint, pairwise homotopic, nonseparating (in Σ ) cycles, and G contains at least ⌊( ρ −1)/8⌋−1 pairwise disjoint, pairwise homotopic, separating, noncontractible cycles.

Three-Coloring Graphs Embedded on Surfaces with All Faces Even-Sided

Every graph embedded on a surface of positive genus with every face bounded by an even number of edges can be 3-colored provided all noncontractible cycles in the graph are sufficiently long. The bound of three colors is the smallest possible for this type of result.

Almost all rooted maps have large representativity

Let M be a map on a surface S. The edge-width of M is the length of a shortest noncontractible cycle of M. The face-width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M. (The edge-width and face-width of a planar map may be defined to be infinity.) A map is a large-edge-width embedding (LEW-embedding) if its maximum face valency is less than its edge-width. For several families of rooted maps on a given surface, we prove that there are positive constants C1 and C2, depending on the family and the surface, such that 1 almost all maps with n edges have face-width and edge-width greater than c1 log n, and 2 the fraction of such maps that are LEW-embeddings and the fraction that are not LEW-embeddings both exceed n− >C2.