Related papers: On the smallest parallel quadrangulation with mini…
In this paper, we introduce a partial order on neighborhood equivalence classes of maximally spread essential multibranched surfaces embedded in a 3-manifold. We show that if a maximally spread essential multibranched surface is atoroidal…
We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.
In this paper we give full classification of rank 3 line arrangements in $\mathbb P^2$ (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to…
We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic…
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the…
We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper…
A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half…
We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…
The notion of noncrossing partitions of a partially ordered set (poset) is introduced here. When the poset in question is $[n]=\{1,2,\dots, n\}$ with the complete order of natural numbers, conventional noncrossing partitions arise. The…
We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs…
A graph is K_{2,3}-saturated if it has no subgraph isomorphic to K_{2,3}, but does contain a K_{2,3} after the addition of any new edge. We prove that the minimum number of edges in a K_{2,3}-saturated graph on n >= 5 vertices is sat(n,…
A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in…
We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree…
A graph in a certain graph class is called minimizing if the least eigenvalue of the adjacency matrix of the graph attains the minimum among all graphs in that class. Bell {\it et al.} have characterized the minimizing graphs in the class…
A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$…
The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted…
We study several properties of $\ZZ_2^n$-equivariant triangulations of $\RR P^n$. We show that a $\ZZ_2^n$-equivariant triangulation of $\RR P^n$ induces a triangulated subdivision of the orbit space $\bigtriangleup^n$. We show that any…
A $k$-cycle in a graph is a cycle of length $k.$ A graph $G$ of order $n$ is called edge-pancyclic if for every integer $k$ with $3\le k\le n,$ every edge of $G$ lies in a $k$-cycle. It seems difficult to determine the minimum size $f(n)$…
We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer.…
We show that 3^7 is the smallest order of a non-trivial odd order group which occurs as the full automorphism group of a finite group.