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Register a new userOn the size of planar graphs with positive Lin-Lu-Yau Ricci curvature
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We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $Delta(G)leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar [{em Trans. Amer. Math. Soc., 2007}] on the size of planar graphs with positive combinatorial curvature.
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In this paper we study the class of compact Kahler manifolds with positive orthogonal Ricci curvature: $Ric^perp>0$. First we illustrate examples of Kahler manifolds with $Ric^perp>0$ on Kahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of Kahler manifolds which admit metrics of $Ric^perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition $Ric^perp>0$ to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four.
For $k ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $Sigma^{4k-1}$ such that $M sharp Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.
In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamiltons classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelmans second paper on the Ricci flow.
A graph $G$ is emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $ein E(G)$. Melnikov and Steinberg [L. S. Melnikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with $n$ vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if $G$ is such a graph with $n(geq6)$ vertices, then $|E(G)|leq frac{5}{2}n-6 $, which improves the upper bound $frac{8}{3}n-frac{17}{3}$ given by Matsumoto [N. Matsumoto, The size of edge-critical uniquely 3-colorable planar graphs, Electron. J. Combin. 20 (3) (2013) $#$P49]. Furthermore, we find some edge-critical 3-colorable planar graphs which have $n(=10,12, 14)$ vertices and $frac{5}{2}n-7$ edges.
Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one positive Ricci curvature is equivalent to positive Ricci curvature and n-positive Ricci curvature is equivalent to positive scalar curvature. Let G be the fundamental group of the closed manifold M. We say that G is virtually free if G contains a free subgroup of finite index, or equivalently, if some finite cover of M has a fundamental group that is a free group. In this paper we will prove: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, such that (n-1)-eigenvalues of the Ricci curvature are strictly positive. Then the fundamental group of M is virtually free. As an immediate consequence we have: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, with 2-positive Ricci curvature. Then the fundamental group of M is virtually free.
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