Papers › Upper bound for the number of spanning forests of regular graphs
Upper bound for the number of spanning forests of regular graphs
Ferenc Bencs, Péter Csikvári
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We show that if G is a d--regular graph on n vertices, then the number of spanning forests F(G) satisfies F(G)≤dⁿ. The previous best bound due to Kahale and Schulman gave (d+1/2+O(1/d))ⁿ. We also have the more precise conjecture that F(G)^(1/n)≤((d-1)ᵈ⁻¹)/(d²-2d-1)^(d/2-1). If this conjecture is true, then the expression on the right hand side is the best possible.
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