Related papers: Compact minimal vertical graphs with non-connected…

We show that constant mean curvature hypersurfaces in $\mathbb H^n\times\mathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces…

Let k>0 be an integer, let H be a minor-minimal graph in the projective plane such that every homotopically non-trivial closed curve intersects H at least k times, and let G be the planar double cover of H obtained by lifting G into the…

Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial…

In this paper we characterise the graphs containing a $\mathbb{Z} \times \mathbb{Z}$ grid minor in a similar way as it has been done by Halin for graphs with an $\mathbb{N} \times \mathbb{Z}$ grid minor. Using our characterisation, we…

We construct for all $ k\in \mathbb{N} $ a $ k $-edge-connected digraph $ D $ with $ s,t\in V(D) $ such that there are no edge-disjoint $ s \rightarrow t $ and $t\rightarrow s $ paths. We use in our construction "self-similar" graphs which…

A graph is hypohamiltonian if it is non-Hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay,…

The existence of Hamiltonian cycles in 1-planar graphs with higher connectivity has attracted considerable attention. Recently, the authors and Dong proved that 4-connected 1-planar chordal graphs are Hamiltonian-connected. In this paper,…

We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph,…

Guo and Royle recently classified the connected cubic graphs without eigenvalues of their adjacency matrix in the open interval $(-1, 1)$, and raised the question of extending their classification to graphs of maximum degree at most $3$.…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical…

We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary.…

Strict outerconfluent drawing is a style of graph drawing in which vertices are drawn on the boundary of a disk, adjacencies are indicated by the existence of smooth curves through a system of tracks within the disk, and no two adjacent…

In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by…

We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many graph classes for which we can compute the diameter in truly subquadratic-time. In particular for any fixed $H$, the class of…

The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic $2$-groups. We show that for any natural $n\ge 2$ there is an undirected graph…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of $H$-free split graphs whose clique-width is…

Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Tur\'an numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $\lambda_k \ge k^{-2k^2}$ such that for any…

We study the class of all finite directed graphs up to primitive positive constructability. The resulting order has a unique greatest element, namely the graph $P_1$ with one vertex and no edges. The graph $P_1$ has a unique greatest lower…

Consider the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the field of $q$ elements ($1<k<n-1$) and denote by $\Pi(n,k)_q$ the restriction of this graph to the set of projective $[n,k]_q$…