Jernej Azarija

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If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd( ↽Ð f ) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube(More)
The Fibonacci cube Γn is obtained from the n-cube Qn by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube Λn is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the(More)
The generalized Fibonacci cube Qd(f) is the subgraph of the d-cube Qd induced on the set of all strings of length d that do not contain f as a substring. It is proved that if Qd(f) ∼= Qd(f ′) then |f | = |f ′|. The key tool to prove this result is a result of Guibas and Odlyzko about the autocorrelation polynomial associated to a binary string. An example(More)
The Fibonacci cube Γn is the subgraph of the n-dimensional cube Qn induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for γt(Γn), n ≤ 12. Consequently, γt(Γn) ≤ 2Fn−10 + 21Fn−8 holds for n ≥ 11, where Fn are the Fibonacci numbers. It is proved that if n ≥ 9, then γt(Γn) ≥ d(Fn+2 − 11)/(n−(More)
Using hypergraph transversals it is proved that γt(Qn+1) = 2γ(Qn), where γt(G) and γ(G) denote the total domination number and the domination number of G, respectively, and Qn is the n-dimensional hypercube. More generally, it is shown that if G is a bipartite graph, then γt(G K2) = 2γ(G). Further, we show that the bipartiteness condition is essential by(More)