173 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The hitting time of clique factors
In a recent paper, Kahn gave the strongest possible, affirmative, answer to
Shamir's problem, which had been open since the late 1970s: Let and
let be divisible by . Then, in the random -uniform hypergraph process
on vertices, as soon as the last isolated vertex disappears, a perfect
matching emerges. In the present work, we transfer this hitting time result to
the setting of clique factors in the random graph process: At the time that the
last vertex joins a copy of the complete graph , the random graph process
contains a -factor. Our proof draws on a novel sequence of couplings,
extending techniques of Riordan and the first author. An analogous result is
proved for clique factors in the -uniform hypergraph process ()
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Paths and cycles in graphs and hypergraphs
In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles.
A -uniform tight cycle is a -uniform hypergraph on vertices with a cyclic ordering of its vertices such that the edges are all -sets of consecutive vertices in the ordering.
We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to -uniform hypergraphs and prove results in the 4- and 5-uniform case.
For a -uniform hypergraph~, the Ramsey number is the smallest integer such that any 2-edge-colouring of the complete -uniform hypergraph on vertices contains a monochromatic copy of . We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that = (5+(1)).
We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any >0 and 3 asymptotically almost surely, every subgraph of the binomial random -uniform hypergraph in which all -sets are contained in at least edges has a tight Hamilton cycle.
A random graph model on a host graph is said to be 1-independent if for every pair of vertex-disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . We show that = 4 - 2 is the critical probability such that every 1-independent graph model on where each edge is present with probability at least contains an infinite path
Large -tilings in 3-uniform hypergraphs
Let be the 3-graph with two edges intersecting in two vertices. We
prove that every 3-graph on vertices with at least
edges contains a -tiling covering more than vertices, for
sufficiently large and . The bound on the number of edges
is asymptotically best possible and solves a conjecture of the authors for
3-graphs that generalizes the Matching Conjecture of Erd\H{o}s
MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs
Constructing null models to test the significance of extracted information is
a crucial step in data analysis. In this work, we provide a uniformly
sampleable null model of directed graphs with the same (or similar) number of
simplices in the flag complex, with the restriction of retaining the underlying
undirected graph. We describe an MCMC-based algorithm to sample from this null
model and statistically investigate the mixing behaviour. This is paired with a
high-performance, Rust-based, publicly available implementation. The motivation
comes from topological data analysis of connectomes in neuroscience. In
particular, we answer the fundamental question: are the high Betti numbers
observed in the investigated graphs evidence of an interesting topology, or are
they merely a byproduct of the high numbers of simplices? Indeed, by applying
our new tool on the connectome of C. Elegans and parts of the statistical
reconstructions of the Blue Brain Project, we find that the Betti numbers
observed are considerable statistical outliers with respect to this new null
model. We thus, for the first time, statistically confirm that topological data
analysis in microscale connectome research is extracting statistically
meaningful information
Transversals via regularity
Given graphs all on the same vertex set and a graph with
, a copy of is transversal or rainbow if it contains at most
one edge from each . When , such a copy contains exactly one edge
from each . We study the case when is spanning and explore how the
regularity blow-up method, that has been so successful in the uncoloured
setting, can be used to find transversals. We provide the analogues of the
tools required to apply this method in the transversal setting. Our main result
is a blow-up lemma for transversals that applies to separable bounded degree
graphs .
Our proofs use weak regularity in the -uniform hypergraph whose edges are
those where is an edge in the graph . We apply our lemma to
give a large class of spanning -uniform linear hypergraphs such that any
sufficiently large uniformly dense -vertex -uniform hypergraph with
minimum vertex degree contains as a subhypergraph. This
extends work of Lenz, Mubayi and Mycroft
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Resilience for Loose Hamilton Cycles
We study the emergence of loose Hamilton cycles in subgraphs of random
hypergraphs. Our main result states that the minimum -degree threshold for
loose Hamiltonicity relative to the random -uniform hypergraph
coincides with its dense analogue whenever . The
value of is approximately tight for . This is particularly
interesting because the dense threshold itself is not known beyond the cases
when .Comment: 33 pages, 3 figure
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