相关论文: A short proof of the existence of a minor-universa…

Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We…

Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). J\'anos Pach (1981) answered this question in the negative. We…

A subgraph-universal graph/a topological minor-universal graph in a class of graphs $\mathcal{G}$ is a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as a subgraph/topological minor. We prove that the class $\mathcal{P}$…

It was proved by Huynh, Mohar, \v{S}\'amal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for…

We give a new proof of the existence of designs, which is much shorter and gives better bounds.

We present an elementary construction of an uncountably chromatic graph without uncountable, infinitely connected subgraphs.

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations.

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an…

Let k>0 be an integer, let H be a minor-minimal graph in the projective plane such that every homotopically non-trivial closed curve intersects H at least k times, and let G be the planar double cover of H obtained by lifting G into the…

In this note, we give short inductive proofs of two known results on $k$-extendible graphs based on a property proved in [Qinglin Yu, A note on $n$-extendable graphs. Journal of Graph Theory, 16:349-353, 1992].

We show that the existence of a universal countably chromatic graph of size $\aleph_1$ together with the failure of continuum hypothesis is consistent. The proof is a forcing iteration of strongly proper ccc posets. The construction works…

We survey algorithms and bounds for constructing planar drawings of graphs in small area.

We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the…

A one-line proof of a minimax theorem due to Steinerberger is given.

We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…

In this paper, we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing. A drawing of a graph is min-$k$-planar if, for every crossing in the drawing, at least one of the two crossing edges involves at…

We give a new short proof for the uniqueness of the universal minimal space. The proof holds for the uniqueness of the universal object in every collection of topological dynamical systems closed under taking projective limits and…

We present a method for constructing countable models of small theories and apply it to prove theorems on the maximal number of countable non-isomorphic models of linearly ordered theories.

An Eulerian-minor of an Eulerian graph is obtained from an Eulerian subgraph of the Eulerian graph by contraction. The Eulerian-minor operation preserves Eulerian properties of graphs and moreover Eulerian graphs are well-quasi-ordered…