Related papers: Universal planar graphs for the topological minor …
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 $\mathcal{P}$…
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…
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 produce a new, shorter construction of a minor-universal planar graph.
Let $W$ be any wheel graph and $\mathcal{G}$ the class of all countable graphs not containing $W$ as a minor. We show that there exists a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as an induced subgraph.
A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable…
We investigate the question how `small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of…
It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end…
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…
A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for…
A graph $G$ is said to be ubiquitous, if every graph $\Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every…
Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden…
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.
Let G be a finite simple graph with automorphism group A(G). Then a spanning subgraph U of G is a fixing subgraph of G if G contains exactly $| A(G)|/ | A(G) \cap A(U)| $ subgraphs isomorphic to U: the graph G must always contain at least…
We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide…
A graph $H$ is ubiquitous if for every graph $G$ that for every natural number $n$ contains $n$ vertex-disjoint $H$-minors contains infinitely many vertex-disjoint $H$-minors. Andreae conjectured that every locally finite graph is…
A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…
For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…
Grohe and Marx proved that if G does not contain H as a topological minor, then there exist constants g=O(|V(H)|^4), D and t depending only on H such that G is a clique sum of graphs that either contain at most t vertices of degree greater…
Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$…