Construction and characterization of graphs whose each spanning tree has a perfect matching
Abstract
An edge subset $S$ of a connected graph $G$ is called an antiKekulé set if $GS$ is connected and has no perfect matching. We can see that a connected graph $G$ has no antiKekulé set if and only if each spanning tree of $G$ has a perfect matching. In this paper, by applying Tutte's 1factor theorem and structure of minimally 2connected graphs, we characterize all graphs whose 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 $n\geq 4$ and size $m$ whose each spanning tree has a perfect matching, then $m\leq \frac{(n+1)n} 2$, with equality if and only if $G\cong K_n\circ K_1$.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1602.00196
 Bibcode:
 2016arXiv160200196W
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 11 pages