Minimum degree and density of binary sequences

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