Randomly perturbing a graph with O(n) random edges forces a colour-biased Hamilton cycle, and at the critical minimum degree the bias is proportional to m.
Colour-bias perfect matchings in hypergraphs
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abstract
We study conditions under which an edge-coloured hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock and Sgueglia, and of Balogh, Treglown and Z\'arate-Guer\'en.
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math.CO 1years
2025 1verdicts
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Colour-biased Hamilton cycles in randomly perturbed graphs
Randomly perturbing a graph with O(n) random edges forces a colour-biased Hamilton cycle, and at the critical minimum degree the bias is proportional to m.