Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs
Abstract
Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary $k$-regular bipartite graphs $G$ on $2n$ vertices for all $k = \omega \left( \frac{n}{\log^{1/3} n} \right)$. Surprisingly, this is not the case for smaller values of $k$. Using a construction due to Goel, Kapralov and Khanna, we show that there exist bipartite $k$-regular graphs in which the last isolated vertex disappears long before a perfect matching appears.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2018
- DOI:
- 10.48550/arXiv.1805.06944
- arXiv:
- arXiv:1805.06944
- Bibcode:
- 2018arXiv180506944G
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- Corrected typo