Abstract
The Fibonacci cube Γ n is the subgraph of the n -dimensional cube Q n induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for γ t (Γ n ) , n ≤ 12 . Consequently, γ t (Γ n ) ≤ 2 F n − 10 + 21 F n − 8 holds for n ≥ 11 , where F n are the Fibonacci numbers. It is proved that if n ≥ 9 , then γ t (Γ n ) ≥ ⌈( F n + 2 −11)/( n −3)⌉ − 1 . Using integer linear programming exact values for the 2-packing number, connected domination number, paired domination number, and signed domination number of small Fibonacci cubes and hypercubes are obtained. A conjecture on the total domination number of hypercubes asserting that γ t ( Q n )=2 n − 2 holds for n ≥ 6 is also disproved in several ways.
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