Graphs where each spanning tree has a perfect matching
DOI:
https://doi.org/10.11575/cdm.v16i3.62279Abstract
An edge subset S of a connected graph G is called an anti-Kekul\'{e} set if G−S is connected and has no perfect matching. We can see that a connected graph G has no anti-Kekul\'{e} set if and only if each spanning tree of G has a perfect matching. In this note, we characterize all graphs where each spanning tree has a perfect matching. In addition, we show that if G is a connected graph of order 2n for a positive integer n≥4 and size m whose each spanning tree has a perfect matching, then m≤(n+1)n/2, with equality if and only if G≅Kn∘K1.
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