相关论文: Minimal flag triangulations of lower-dimensional m…

In this survey article, we are interested on minimal triangulations of closed pl manifolds. We present a brief survey on the works done in last 25 years on the following: (i) Finding the minimal number of vertices required to triangulate a…

We present small triangulations of all connected sums of $\mathbb{CP}^2$ and $S^2 \times S^2$ with the standard piecewise linear structure. Our triangulations have $2\beta_2+2$ pentachora, where $\beta_2$ is the second Betti number of the…

In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$…

We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb S^{d-2}$-bundles over $\mathbb S^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is non-orientable, the minimum…

We consider the problem of minimising the number of edges that are contained in triangles, among $n$-vertex graphs with a given number of edges. We prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum, for…

We study lower bounds for the number of vertices in a PL-triangulation of a given manifold $M$. While most of the previous estimates are based on the dimension and the connectivity of $M$, we show that further information can be extracted…

We prove that among all flag 3-manifolds on $n$ vertices, the join of two circles with $\left\lceil{\frac{n}{2}}\right \rceil$ and $\left\lfloor{\frac{n}{2}}\right \rfloor$ vertices respectively is the unique maximizer of the face numbers.…

We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $\Delta$, we show that there exists a balanced complex with the same $f$-vector as $\Delta$, and whose top-dimensional Betti number is…

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}P^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated…

We characterize f-vectors of sufficiently large three-dimensional flag Gorenstein* complexes, essentially confirming a conjecture of Gal [Discrete Comput. Geom., 34 (2), 269--284, 2005]. In particular, this characterizes f-vectors of large…

Let $\Delta$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $\Delta$ is that $g_2$ of $\Delta$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $\Delta$ has…

A recent conjecture of Chudnovsky and Nevo asserts that flag triangulations of spheres always have linear-sized independent sets, with a precisely conjectured proportion depending on the dimension. For dimensions one and two, the lower…

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed…

A planar graph $G$ is called a pentagulation of an $n$-gon ($n\geq$ is an integer) if all faces of $G$ are pentagons, except one, which is an $n$-gon. A $3$-connected pentagulation $G$ of an $n$-gon is called minimal if it has the smallest…

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is…

A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a…

Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool…

We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold…

We prove that every locally Hamiltonian graph with $n\ge 3$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a…

Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the…