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Blow-up Lemma
Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
Proof of the Seymour conjecture for large graphs
Paul Seymour conjectured that any graphG of ordern and minimum degree of at leastk/k+1n contains thekth power of a Hamiltonian cycle. Here, we prove this conjecture for sufficiently largen.
On the Pósa-Seymour conjecture
TLDR
The following approximate version of Paul Seymour's conjecture is proved: for any > 0 and positive integer k, there is an n0 such that, if G has order n ≥ n0 and minimum degree at least ( k k+1 + )n, then G contains the kth power of a Hamilton cycle.
Three-Color Ramsey Numbers For Paths
TLDR
It is proved that for sufficiently large n, for the three-color Ramsey numbers of paths on n vertices, the following conjecture of Faudree and Schelp is proved.
An algorithmic version of the blow-up lemma
TLDR
This paper provides an algorithmic version of the Blow-up Lemma, which helps to find bounded degree spanning subgraphs in $\varepsilon$-regular graphs and can be parallelized and implemented in $NC^5$.
Proof of a Packing Conjecture of Bollobás
TLDR
Here it is proved that for any positive integer Δ and real 0 c n 0, T is a tree of order n and maximum degree Δ, and G is a graph ofOrder n andmaximum degree not exceeding cn , then there is a packing of T and G.
Ramsey‐type results for Gallai colorings
TLDR
It is proved that in every G-coloring of Kn there exists each of the following: a monochromatic double star with at least 3n+1 4 vertices; and RG(r,K3) can be determined exactly.
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