107 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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
Optimal Hamilton covers and linear arboricity for random graphs
In his seminal 1976 paper, P\'osa showed that for all , the
binomial random graph is with high probability Hamiltonian. This leads
to the following natural questions, which have been extensively studied: How
well is it typically possible to cover all edges of with Hamilton
cycles? How many cycles are necessary? In this paper we show that for , we can cover with precisely
Hamilton cycles. Our result is clearly best possible
both in terms of the number of required cycles, and the asymptotics of the edge
probability , since it starts working at the weak threshold needed for
Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and
improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of
Ferber, Kronenberg and Long, essentially closing a long line of research on
Hamiltonian packing and covering problems in random graphs.Comment: 13 page
Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture
An orthomorphism of a finite group is a bijection
such that is also a bijection. In 1981, Friedlander,
Gordon, and Tannenbaum conjectured that when is abelian, for any
dividing , there exists an orthomorphism of fixing the identity and
permuting the remaining elements as products of disjoint -cycles. We prove
this conjecture for all sufficiently large groups.Comment: 34 page
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture
In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any
-vertex graph can be decomposed into cycles and edges. We improve
upon the previous best bound of cycles and edges due to
Conlon, Fox and Sudakov, by showing an -vertex graph can always be
decomposed into cycles and edges, where is the
iterated logarithm function.Comment: Final version, accepted for publicatio
A random Hall-Paige conjecture
A complete mapping of a group is a bijection such
that is also bijective. Hall and Paige conjectured in 1955
that a finite group has a complete mapping whenever is
the identity in the abelianization of . This was confirmed in 2009 by
Wilcox, Evans, and Bray with a proof using the classification of finite simple
groups. In this paper, we give a combinatorial proof of a far-reaching
generalisation of the Hall-Paige conjecture for large groups. We show that for
random-like and equal-sized subsets of a group , there exists a
bijection such that is a bijection from
to whenever in the
abelianization of . Using this result, we settle the following conjectures
for sufficiently large groups. (1) We confirm in a strong form a conjecture of
Snevily by characterising large subsquares of multiplication tables of finite
groups that admit transversals. Previously, this characterisation was known
only for abelian groups of odd order. (2) We characterise the abelian groups
that can be partitioned into zero-sum sets of specific sizes, solving a problem
of Tannenbaum, and confirming a conjecture of Cichacz. (3) We characterise
harmonious groups, that is, groups with an ordering in which the product of
each consecutive pair of elements is distinct, solving a problem of Evans. (4)
We characterise the groups with which any path can be assigned a cordial
labelling. In the case of abelian groups, this confirms a conjecture of Patrias
and Pechenik
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