arXiv subjects

Max Pitz

Publications and source records attributed to Max Pitz.

50 records · Page 3Linked to original sources

arXiv2018

Partitioning edge-coloured infinite complete bipartite graphs into monochromatic paths

In 1978, Richard Rado showed that every edge-coloured complete graph of countably infinite order can be partitioned into monochromatic paths of different colours. He asked whether this remains true for uncountable complete graphs and a notion of \emph{generalised paths}. In 2016, Daniel Soukup answered this in the affirmative and conjectured that a similar result should hold for complete bipartite graphs with bipartition classes of the same infinite cardinality, namely that every such graph edge-coloured with $r$ colours can be partitioned into $2r-1$ monochromatic generalised paths with each colour being used at most twice. In the present paper, we give an affirmative answer to Soukup's conjecture.

math.CO↗

arXiv2018

$n$-arc and $n$-circle connected graph-like spaces

A space $X$ is $n$-arc connected (respectively, $n$-circle connected) if for any choice of at most $n$ points there is an arc (respectively, a circle) in $X$ containing the specified points. We study $n$-arc connectedness and $n$-circle connectedness in compactifications of locally finite graphs and the slightly more general class of graph-like continua, uncovering a striking difference in their behaviour regarding $n$-arc and -circle connectedness.

math.CO↗

arXiv2018

Ubiquity in graphs I: Topological ubiquity of trees

Let $\triangleleft$ be a relation between graphs. We say a graph $G$ is \emph{$\triangleleft$-ubiquitous} if whenever $Γ$ is a graph with $nG \triangleleft Γ$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \triangleleft Γ$, where $αG$ is the disjoint union of $α$ many copies of $G$. The \emph{Ubiquity Conjecture} of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper, which is the first of a series of papers making progress towards the Ubiquity Conjecture, we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.

math.CO↗

arXiv2018

n-Arc Connected Graphs

Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G| the one-complex associated with $G$. The topological space |G| is n-arc connected (n-ac) if every set of no more than n points of |G| are contained in an arc (a homeomorphic copy of the closed unit interval). For any graph G, we show the following are equivalent: (i) |G| in 7-ac, (ii) |G| is n-ac for all n, and (iii) G is a subdivision of one of nine graphs. A graph G has |G| 6-ac if and only if either G is one of the nine 7-ac graphs, or, after suppressing all degree-2-vertices, the graph G is 3-regular, 3-connected, and removing any 6 edges does not disconnect G into 4 or more components. Similar combinatorial characterizations of graphs G such that |G| is n-ac for n=3, 4 and 5 are given. Together these results yield a complete classification of n-ac graphs, for all n.

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arXiv2018

Partitioning Edge-Coloured Complete Symmetric Digraphs into Monochromatic Complete Subgraphs

Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$. However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every $(r+1)$-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochromatic complete symmetric digraphs in colour $r+1$. Furthermore, we present a stability version for the countable case of the latter result: We prove that the edge-colouring is uniquely determined on a large subgraph, as soon as the upper density of monochromatic paths in colour $r+1$ is bounded by $\prod_{i\in [r]}\frac{1}{\ell_i}$.

math.CO↗

arXiv2018

Ends, tangles and critical vertex sets

We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_Γ$ is a quotient of Diestel's (denoted by $\vert G\vert_Θ$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of $\vert G\vert_Θ$ and our construction of $\vert G\vert_Γ$, we show that $G$ can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's $\vert G\vert_Θ$ is the finest such compactification, and our $\vert G\vert_Γ$ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.

math.CO↗

arXiv2018

Non-reconstructible locally finite graphs

Two graphs $G$ and $H$ are \emph{hypomorphic} if there exists a bijection $φ\colon V(G) \rightarrow V(H)$ such that $G - v \cong H - φ(v)$ for each $v \in V(G)$. A graph $G$ is \emph{reconstructible} if $H \cong G$ for all $H$ hypomorphic to $G$. Nash-Williams proved that all locally finite graphs with a finite number $\geq 2$ of ends are reconstructible, and asked whether locally finite graphs with one end or countably many ends are also reconstructible. In this paper we construct non-reconstructible graphs of bounded maximum degree with one and countably many ends respectively, answering the two questions of Nash-Williams about the reconstruction of locally finite graphs in the negative.

math.CO↗

arXiv2018

A counterexample to the reconstruction conjecture for locally finite trees

Two graphs $G$ and $H$ are hypomorphic if there exists a bijection $φ\colon V(G) \rightarrow V(H)$ such that $G - v \cong H - φ(v)$ for each $v \in V(G)$. A graph $G$ is reconstructible if $H \cong G$ for all $H$ hypomorphic to $G$. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we construct a counterexample to the Harary-Schwenk-Scott Conjecture. Our example also answers four other questions of Nash-Williams, Halin and Andreae on the reconstruction of infinite graphs.

math.CO↗

arXiv2017

Decomposing edge-coloured complete symmetric digraphs into monochromatic paths

Confirming and extending a conjecture by Guggiari, we show that every countable $(r+1)$-edge-coloured complete symmetric digraph containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochromatic directed paths in colour $r+1$.

math.CO↗

arXiv2017

Minimal obstructions for normal spanning trees

Diestel and Leader have characterised connected graphs that admit a normal spanning tree via two classes of forbidden minors. One class are Halin's $(\aleph_0,\aleph_1)$-graphs: bipartite graphs with bipartition $(\mathbb{N},B)$ such that $B$ is uncountable and every vertex of $B$ has infinite degree. Our main result is that under Martin's Axiom and the failure of the Continuum Hypothesis, the class of forbidden $(\aleph_0,\aleph_1)$-graphs in Diestel and Leader's result can be replaced by one single instance of such a graph. Under CH, however, the class of $(\aleph_0,\aleph_1)$-graphs contains minor-incomparable elements, namely graphs of binary type, and $\mathcal{U}$-indivisible graphs. Assuming CH, Diestel and Leader asked whether every $(\aleph_0,\aleph_1)$-graph has an $(\aleph_0,\aleph_1)$-minor that is either indivisible or of binary type, and whether any two $\mathcal{U}$-indivisible graphs are necessarily minors of each other. For both questions, we construct examples showing that the answer is in the negative.

math.CO↗

arXiv2017

Hamilton decompositions of one-ended Cayley graphs

We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a decomposition into $n$ edge-disjoint Hamiltonian double-rays for all $n \in \mathbb{N}$.

math.CO↗

arXiv2017

Hamilton cycles in infinite cubic graphs

Investigating a problem of B. Mohar, we show that every one-ended Hamiltonian cubic graph with end degree 3 contains a second Hamilton cycle. We also construct two examples showing that this result does not extend to give a third Hamilton cycle, nor that it extends to the two-ended case.

math.CO↗

arXiv2017

A counterexample to Montgomery's conjecture on dynamic colourings of regular graphs

A \emph{dynamic colouring} of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The \emph{dynamic colouring number} $χ_2(G)$ of a graph $G$ is the least number of colours needed for a dynamic colouring of $G$. Montgomery conjectured that $χ_2(G) \leq χ(G) + 2$ for all regular graphs $G$, which would significantly improve the best current upper bound $χ_2(G) \leq 2χ(G)$. In this note, however, we show that this last upper bound is sharp by constructing, for every integer $n \geq 2$, a regular graph $G$ with $χ(G) = n$ but $χ_2(G) = 2n$. In particular, this disproves Montgomery's conjecture.

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arXiv2016

Graph-Like Compacta: Characterizations and Eulerian Loops

A compact graph-like space is a triple $(X,V,E)$ where $X$ is a compact, metrizable space, $V \subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X \setminus V \cong E \times (0,1)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.

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