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.

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.Expand

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.Expand

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 $NC^5$.Expand

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.Expand

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.Expand