6 research outputs found
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
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum