Minimum degree and density of binary sequences

3 For d, k ∈ N with k ≤ 2d, let g(d, k) denote the infimum density of binary 4 sequences (xi)i∈Z ∈ {0, 1}Z which satisfy the minimum degree condition d ∑ j=1 (xi+j + 5 xi−j) ≥ k for all i ∈ Z with xi = 1. We reduce the problem to determine g(d, k) 6 to a combinatorial problem related to the generalized k-girth of a graph G which 7 is defined as the minimum order of an induced subgraph of G of minimum degree 8 at least k. Extending results of Kézdy and Markert, and of Bermond and Peyrat, 9 we present a minimum mean cycle formulation which allows to determine g(d, k) for 10 small values of d and k. For odd values of k with d + 1 ≤ k ≤ 2d, we conjecture 11 g(d, k) = k 2−1 2(dk−1) and show that this holds for k ≥ 2d− 3. 12