||1. A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove that a connected graph $G$ is near-factor-critical if and only if it has a perfect matching. We also characterize disconnected near-factor-critical graphs.
2. The entropy compression is a new developed and inspiring method. And it was widely discussed by combinatorists. For example, see . In this talk, I will describe the idea behind the method. Also, I will introduce the results obtained by entropy compression and sketch their proofs in ,,,. |
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