Max Pitz

A TOPOLOGICAL VARIATION OF THE RECONSTRUCTION CONJECTURE

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals ℝ, the rationals ℚ and the irrationals ℙ are reconstructible, as well as spaces occurring as Stone–Cech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C .

Non-reconstructible locally finite graphs

Two graphs

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

G

A counterexample to the reconstruction conjecture for locally finite trees

and

Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

H

Ubiquity in graphs I: Topological ubiquity of trees

are \emph{hypomorphic} if there exists a bijection

Ends, tangles and critical vertex sets

\varphi \colon V(G) \rightarrow V(H)

The Stone-Cech compactifications of

such that

\omega^*\setminus \{x\}

G - v \cong H - \varphi(v)

and

for each