Joshua Erde

Decomposing the cube into paths

We consider the question of when the n -dimensional hypercube can be decomposed into paths of length k . For odd n it is necessary that k divides n 2 n - 1 and that k ? n . Anick and Ramras (2013) conjectured that these two conditions are also sufficient for all odd n and prove that this is true for odd n ? 2 32 . In this note we prove the conjecture.

Non-reconstructible locally finite graphs

Two graphs

Duality Theorems for Blocks and Tangles in Graphs

G

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

and

A counterexample to the reconstruction conjecture for locally finite trees

H

Isoperimetry in integer lattices

are \emph{hypomorphic} if there exists a bijection

Refining a Tree-Decomposition which Distinguishes Tangles

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

Duality Theorems for Tangles in Graphs

such that

A Note on Combinatorial Derivation

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

The closed knight tour problem in higher dimensions

for each