Related papers: Tiltan and graphs with no infinite paths
In this paper, we study the Tur\'{a}n problem of signed graphs version. Suppose that $\dot{G}$ is a connected unbalanced signed graph of order $n$ with $e(\dot{G})$ edges and $e^-(\dot{G})$ negative edges, and let $\rho(\dot{G})$ be the…
We give a sufficient condition on totally disconnected topological graphs such that their associated topological graph algebras are purely infinite.
We determine the colored patterns that appear in any $2$-edge coloring of $K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We prove the existence of a positive integer $z_2$ such that any $2$-edge coloring of…
In this note, we prove that a finite vertex-transitive graph which has a clique which intersects all maximal cliques is a complete graph. This gives a positive answer in the case of vertex-transitive graphs to a question raised by Berge and…
It was recently proved that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $\omega \geq \frac 23(\Delta+1)$ unless the…
A geometric graph in the plane is angle-monotone of width $\gamma$ if every pair of vertices is connected by an angle-monotone path of width $\gamma$, a path such that the angles of any two edges in the path differ by at most $\gamma$.…
We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…
We detect, by using symplectic topology, invariant measures with large rotation vectors for a class of Hamiltonian flows.
Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along…
To a finite, connected, unoriented graph of Betti-number g>=2 and valencies >=3 we associate a finitely summable, commutative spectral triple (in the sense of Connes), whose induced zeta functions encode the graph. This gives another…
We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above…
We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for…
We prove almost tight bounds on the length of paths in $2$-edge-connected cubic graphs. Concretely, we show that (i) every $2$-edge-connected cubic graph of size $n$ has a path of length $\Omega\left(\frac{\log^2{n}}{\log{\log{n}}}\right)$,…
A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The…
The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$…
It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each…
Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $\sigma: E(G)\rightarrow \{+, -\}$. Denoted by $i_+(\Gamma)$, $\theta(\Gamma)$ and $p(\Gamma)$ the positive inertia index, the cyclomatic…
We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~$C^{1,\gamma}$. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.
Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the…
We obtain necessary and sufficient conditions for pure infiniteness of the path groupoid $C^*$-algebra of a row-finite graph without sinks. In particular we show that for such a path groupoid $\mathcal{G}_E$, the properties of being…