On the Upper Bounds of the Numbers of Perfect Matchings in Graphs with Given Parameters

Abstract

Let ϕ(G), κ(G), α(G), χ(G), cl(G), diam(G) denote the number of perfect matchings, connectivity, independence number, chromatic number, clique number and diameter of a graph G, respectively. In this note, by constructing some extremal graphs, the following extremal problems are solved:

1. max {ϕ(G): |V (G)| = 2n, κ(G) ≤ k} = k[(2n − 3)!!],

2. $\max {\left\{ {\phi {\left( G \right)}:{\left| {V{\left( G \right)}} \right|} = 2n,\alpha {\left( G \right)} \geqslant k} \right\}} = {\left[ {{\prod\limits_{i = 0}^{k - 1} {{\left( {2n - k - i} \right)}} }} \right]}{\left[ {{\left( {2n - 2k - 1} \right)}!!} \right]},$

3. max{ϕ(G): |V (G)| = 2n, χ(G) ≤ k} = ϕ(T _k,2n ) T _k,2n is the Turán graph, that is a complete k-partite graph on 2n vertices in which all parts are as equal in size as possible,

4. max{ϕ(G): |V (G)| = 2n, cl(G) = 2} = n!,

5. max{ϕ(G): |V (G)| = 2n, diam(G) ≥ 2} = (2n − 2)(2n − 3)[(2n − 5)!!],

max{ϕ(G): |V (G)| = 2n, diam(G) ≥ 3} = (n − 1)²[(2n − 5)!!].