Related papers: Lattices with many congruences are planar
A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let $F_n$ be a fullerene graph with $n$ vertices. A set $\mathcal H$ of mutually disjoint hexagons of $F_n$ is a sextet pattern if…
For a slim, planar, semimodular lattice $L$ and covering square~$S$, G.~Cz\'edli and E.\,T.~Schmidt introduced the fork extension, $L[S]$, which is also a slim, planar, semimodular lattice. We investigate when a congruence of $L$ extends to…
A 2-nearly Platonic graph of type (k|d) is a k-regular planar graph with f faces, f-2 of which are of degree d and the remaining two are of degrees m_1;m_2, both different from d. Such a graph is called balanced if m_1=m_2. We show that all…
We prove a conjecture of Dvo\v{r}\'ak, Kr\'al, Nejedl\'y, and \v{S}krekovski that planar graphs of girth at least five are square $(\Delta+2)$-colorable for large enough $\Delta$. In fact, we prove the stronger statement that such graphs…
A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that…
We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of…
We characterize the order of principal congruences of a bounded lattice as a bounded ordered set. We also state a number of open problems in this new field.
We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear growth of these equations. We then use these recurrences to search for lattice equations that have linear growth…
Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in a linearly ordered set. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions of…
A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a…
We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the…
A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…
A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a…
A graph is called $1$-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let $G$ be a bipartite 1-planar graph with $n$ ($\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for even…
A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph…
The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in $\mathbb R^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it…
The Clar number of a fullerene is the maximum number of independent resonant hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by [n/6]-2. We find that there are no fullerenes whose…
We prove that a tolerance relation of a lattice is a homomorphic image of a congruence relation.
An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. We prove for $r \in \{4,5\}$, every projective planar $r$-graph with no Petersen-minor is $r$-edge colorable.
A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of…