Related papers: A bijection for triangulations, quadrangulations, …
A particular case of Caccetta-H\"{a}ggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order…
Generalized Tamari intervals have been recently introduced by Pr\'eville-Ratelle and Viennot, and have been proved to be in bijection with (rooted planar) non-separable maps by Fang and Pr\'eville-Ratelle. We present two new bijections…
We consider large random planar maps and study the first-passage percolation distance obtained by assigning independent identically distributed lengths to the edges. We consider the cases of quadrangulations and of general planar maps. In…
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…
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more…
Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we…
We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics…
We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the…
In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this…
The triangulations of a regular convex polygon are enumerated according to the number of diagonals parallel to a fixed edge. The enumeration uses the Shapiro convolution identity, as well as an interpretation of this identity in terms of…
The main objects of the paper are $z$-oriented triangulations of connected closed $2$-dimensional surfaces. A $z$-orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an…
In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that…
We give an explicit construction of vertex-transitive tight triangulations of $d$-manifolds for $d\geq 2$. More explicitly, for each $d\geq 2$, we construct two $(d^2+5d+5)$-vertex neighborly triangulated $d$-manifolds whose vertex-links…
This paper studies the minimal number of vertices $\lambda(n,d)$ required in a triangulation of the $n$-sphere to admit a simplicial map to the boundary of a $(n+1)$-simplex with a given degree $d$. We establish upper bounds for…
We say two posets are "doppelg\"angers" if they have the same number of $P$-partitions of each height $k$. We give a uniform framework for bijective proofs that posets are doppelg\"angers by synthesizing $K$-theoretic Schubert calculus…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair…
A matching cut is a partition of the vertex set of a graph into two sets $A$ and $B$ such that each vertex has at most one neighbor in the other side of the cut. The MATCHING CUT problem asks whether a graph has a matching cut, and has been…
We give a combinatorial proof of a recent result of B\'ona by constructing a bijection from the set of all neighbors of leaves of increasing trees of size $n$ to the set of derangements of length $n$.
We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain…