On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.     

On a long-standing conjecture by Pólya–Szegö and related topics

The electrostatic capacity of a convex body is usually not simple to compute. We discuss two possible approximations of it. The first one is related to a long-standing conjecture by Pólya–Szegö. It states that, among all convex bodies, the “worst shape” for the approximation exists and is the planar disk. We prove the first part of this conjecture, and we establish some related results which give further evidence for the validity of the second part. We also suggest some complementary conjectures and open problems. The second approximation we study is based on the use of web functions.Received: September 29, 2003     

On a long-standing conjecture by Pólya–Szeg? and related topics

The electrostatic capacity of a convex body is usually not simple to compute. We discuss two possible approximations of it. The first one is related to a long-standing conjecture by Pólya–Szegö. It states that, among all convex bodies, the “worst shape” for the approximation exists and is the planar disk. We prove the first part of this conjecture, and we establish some related results which give further evidence for the validity of the second part. We also suggest some complementary conjectures and open problems. The second approximation we study is based on the use of web functions.     

Neighbor Sum Distinguishing Total Choice Number of Planar Graphs without 6-cycles

Pil?niak and Wo?niak put forward the concept of neighbor sum distinguishing (NSD) total coloring and conjectured that any graph with maximum degree Δ admits an NSD total (Δ+3)-coloring in 2015. In 2016, Qu et al. showed that the list version of the conjecture holds for any planar graph with Δ ≥ 13. In this paper, we prove that any planar graph with Δ ≥ 7 but without 6-cycles satisfies the list version of the conjecture.     

Cycle Covers of Planar 2-Edge-Connected Graphs

A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n ? 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n ? 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.     

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ?ⁿ. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].     

An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.     

Proof of Ding’s conjecture on maximal stable sets and maximal cliques in planar graphs

X. Deng et al. proved Chvātal＇s conjecture on maximal stable sets and maximal cliques in graphs. G. Ding made a conjecture to generalize Chvátal＇s conjecture. The purpose of this paper is to prove this conjecture in planar graphs and the complement of planar graphs.     

Remarks on the bondage number of planar graphs

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. In 1998, J.E. Dunbar, T.W. Haynes, U. Teschner, and L. Volkmann posed the conjecture b(G)Δ(G)+1 for every nontrivial connected planar graph G. Two years later, L. Kang and J. Yuan proved b(G)8 for every connected planar graph G, and therefore, they confirmed the conjecture for Δ(G)7. In this paper we show that this conjecture is valid for all connected planar graphs of girth g(G)4 and maximum degree Δ(G)5 as well as for all not 3-regular graphs of girth g(G)5. Some further related results and open problems are also presented.     

Matching theory and Barnette's conjecture

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian.A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity.As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. (1985) [14]. These allow us to check whether a graph we generated is a brace.     

Total Curvature and Spiralling Shortest Paths

This paper gives a partial confirmation of a conjecture of Agarwal, Har-Peled, Sharir, and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in R ³ cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an example is constructed to show that the total curvature of a shortest path on the boundary of a convex polyhedron in R ³ can exceed 2. Another example shows that the spiralling number of a shortest path on the boundary of a convex polyhedron can be arbitrarily large.     

A curvature identity on a 6-dimensional Riemannian manifold and its applications

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting.     

Further extensions of the Grötzsch Theorem

The Grötzsch Theorem states that every triangle-free planar graph admits a proper 3-coloring. Among many of its generalizations, the one of Grünbaum and Aksenov, giving 3-colorability of planar graphs with at most three triangles, is perhaps the most known. A lot of attention was also given to extending 3-colorings of subgraphs to the whole graph. In this paper, we consider 3-colorings of planar graphs with at most one triangle. Particularly, we show that precoloring of any two non-adjacent vertices and precoloring of a face of length at most 4 can be extended to a 3-coloring of the graph. Additionally, we show that for every vertex of degree at most 3, a precoloring of its neighborhood with the same color extends to a 3-coloring of the graph. The latter result implies an affirmative answer to a conjecture on adynamic coloring. All the presented results are tight.     

Hyperbolic Carathéodory conjecture

A quadratic point on a surface in ℝP³ is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjecture is similar to the relation between the six-vertex and the four-vertex theorems on plane curves. Examples of quartic perturbations of the standard hyperboloid confirm our conjecture. Our main result is a linearization and reformulation of the problem in the framework of the 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing the Tresse-Wilczynski theorem. Dedicated to Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Maximum Genus of Strong Embeddings

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover.Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.     

A note on Fischer-Marsden's conjecture

In this paper, we borrowed some ideas from general relativity and find a Robinson-type identity for the overdetermined system of partial differential equations in the Fischer-Marsden conjecture. We proved that if there is a nontrivial solution for such an overdetermined system on a 3-dimensional, closed manifold with positive scalar curvature, then the manifold contains a totally geodesic 2-sphere.

     

Cuts in matchings of 3-connected cubic graphs

We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochstättler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochstättler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.     

On a conjecture of Tutte concerning minimal tree numbers

A counterexample is given to a conjecture by Tutte on the minimum number of spanning trees that a 3-connected planar graph with a prescribed number of edges may have.     

Longest cycles and their chords

Thomassen conjectured that any longest cycle of a 3-connected graph has a chord. In this paper, we will show that the conjecture is true for a planar graph if it is cubic or δ ? 4.     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.