Large Hamiltonian graphs with minimum degree n to the power 1 minus a small epsilon contain a 2-factor consisting of exactly k cycles.
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For sufficiently large n with r = floor((n-1)/2), any Hamiltonian Berge cycle plus one additional edge in an n-vertex r-uniform hypergraph contains Berge cycles of all lengths 2 to n.
Proves that graphs on N ≥ 2n vertices with δ(G) ≥ ⌊3N/4⌋ have every 2-edge-coloring containing a monochromatic copy of every n-vertex tree with max degree ≤ Δ.
citing papers explorer
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On $2$-factors of Hamiltonian graphs
Large Hamiltonian graphs with minimum degree n to the power 1 minus a small epsilon contain a 2-factor consisting of exactly k cycles.
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One extra edge forces Berge pancyclicity
For sufficiently large n with r = floor((n-1)/2), any Hamiltonian Berge cycle plus one additional edge in an n-vertex r-uniform hypergraph contains Berge cycles of all lengths 2 to n.
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A degree version of the Burr-Erd\H{o}s conjecture on trees
Proves that graphs on N ≥ 2n vertices with δ(G) ≥ ⌊3N/4⌋ have every 2-edge-coloring containing a monochromatic copy of every n-vertex tree with max degree ≤ Δ.