Related papers: Diagrams and rectangular extensions of planar semi…

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 lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Cz\'edli and E.T. Schmidt as the duals of the lattices…

The systematic study of planar semimodular lattices started in 2007 with a series of papers by G. Gr\"atzer and E. Knapp. These lattices have connections with group theory and geometry. A planar semimodular lattice $L$ is {\it slim} if…

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a…

A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_{3}$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No…

The Gr\"{o}tzsch Theorem states that every triangle-free planar graph admits a proper $3$-coloring. Among many of its generalizations, the one of Gr\"{u}nbaum and Aksenov, giving $3$-colorability of planar graphs with at most three…

A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_3$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No…

Slim semimodular lattices were introduced by G. Gr\"atzer and E. Knapp in 2007, and they have intensively been studied since then. It is often reasonable to give these lattices by their $\mathcal C_1$-diagrams defined by the author in 2017.…

A 1955 result of J.~Jakub\'i k states that for the prime intervals $\fp$ and $\fq$ of a finite lattice, $\con{\fp} \geq \con{\fq}$ if{}f $\fp$ is congruence-projective to~$\fq$ (\emph{via} intervals of arbitrary size). The problem is how to…

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a…

A recent result of G. Cz\'edli and E.\,T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding "forks". We give a construction that accomplishes the same by deleting…

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for "many" objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general framework under which it is possible to…

IIn a finite lattice, a congruence spreads from a prime interval to another by a sequence of congruence-perspectivities through \emph{intervals of arbitrary size}, by a 1955 result of J. Jakub\'ik. In this note, I introduce the concept of…

In 1960, G. Gr\"atzer and E.\,T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice $L$. For $u \leq v$ in $L$, they constructed a sectional…

A classical result of R.\,P. Dilworth states that every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice~$L$. A~sharper form was published in G.~Gr\"atzer and E.\,T. Schmidt in 1962, adding…

A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation…

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of these…

G. Cz\'edli and E.\,T. Schmidt introduced in 2012 the fork extension. Continuing from Part I, we investigate the congruences of a fork extension. This paper has been merged with Part I, under the title Congruences of fork extensions of slim…

We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd cycle $C_{2k+1}$.…

A construction described by the current author in 2017 uses two linear `prototype' graphs to build a compound graph with Ramsey properties inherited from the prototypes. This paper describes a generalisation of that construction which has…