Related papers: Hadwiger's conjecture: finite vs infinite graphs

Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree.…

We study the behaviour of random labelled and unlabelled cographs with n vertices as n tends to infinity. Our main result is a novel probabilistic limit in the space of graphons.

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their…

Halin proved that every graph with an end $\omega$ containing infinitely many pairwise disjoint rays admits a subdivision of the infinite quarter-grid as a subgraph where all rays from that subgraph belong to $\omega$. We will prove a…

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 introduce the following weak version of Hadwiger's conjecture: If $G$ is a graph and $\kappa$ is a cardinal such that there is no coloring map $c:G \to \kappa$, then $K_\kappa$ is a minor of $G$. We prove that this statement is true for…

Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite…

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from…

The asymptotic study of percolation on finite transitive graphs is considered. Several questions and very few answers regarding percolation on finite graphs are presented.

The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) >= \chi(G). Since \chi(G) \alpha(G) >= |V(G)|, Hadwiger's Conjecture implies that \alpha(G) h(G) >= |V(G)|. We show…

Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs.

We prove that there exist graphs which do not contain $K_t$ as an odd minor and whose chromatic number is at least $(\frac 32-o(1))t$. This disproves, in a strong form, the odd Hadwiger conjecture of Gerards and Seymour from 1993.

Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of pseudoharmonic function are obtained.

In finite graphs, finite-order tangles offer an abstract description of highly connected substructures. In infinite graphs, infinite-order tangles compactify the graphs in the same way the ends compactify connected locally finite graphs.…

We prove that every 2k-edge-connected graph with countably many edge-ends admits a k-arc-connected orientation, extending the previous result by Assem, Koloschin and Pitz that also assumed the hypothesis of the graph being locally finite.…

In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines.

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number…

We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck…

We show that the topological space of any infinite graph and its ends is normal. In particular, end spaces themselves are normal.