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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 67 of 349

  • Full rainbow matchings in graphs and hypergraphs

  • Part of
  • Pu Gao, Reshma Ramadurai, Ian M. Wanless, Nick Wormald
  • Journal:
    Combinatorics, Probability and Computing / Volume 30 / Issue 5 / September 2021
    Published online by Cambridge University Press:
    20 January 2021, pp. 762-780

  • Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\], where \[c > 9/10\], and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from Mi for each \[1 \leqslant i \leqslant m\]. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs.

    Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

Page 67 of 349