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Counterexamples to conjectures on strong maximality and minimality
Lawrence Hollom, Benedict Randall Shaw
We provide counterexamples to several conjectures concerning strongly maximal and strongly minimal structures in infinite graphs and hypergraphs. In particular, we construct 3-unif…
Reverse Littlewood--Offord problems with parity conditions
Lawrence Hollom, Gregory B. Sorkin
We consider the probability that the random signed sum lies within a given distance of the origin, where are fixe…
Connecting hypercube 1-factors
Lawrence Hollom, Benedict Randall Shaw
A 1-factorisation of a regular graph is a partition of its edge set into perfect matchings of . Behague asked for the minimal such that some -factorisatio…
Approximate Itai-Zehavi conjecture for random graphs
Lawrence Hollom, Lyuben Lichev, Adva Mond +2
A famous conjecture by Itai and Zehavi states that, for every -vertex-connected graph and every vertex in , there are spanning trees of such that, for every v…
Finding long cycles in a percolated expander graphs
Lawrence Hollom
Given a graph , the percolated graph has each edge independently retained with probability . Collares, Diskin, Erde, and Krivelevich initiated the study of large struct…
Monotonicity and decompositions of random regular graphs
Lawrence Hollom, Lyuben Lichev, Adva Mond +2
In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameter…
Double-jump phase transition for the reverse Littlewood--Offord problem
Lawrence Hollom, Julien Portier, Victor Souza
Erdős conjectured in 1945 that for any unit vectors in and signs taken independently and uniformly in $\{-1…