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A convenient operator representation for zig-zag four-point and two-point planar Feynman diagrams relevant to the bi-scalar D-dimensional “fishnet” field theory is obtained. This representation gives a possibility to evaluate exactly diagrams of the zig-zag series in special cases. In particular, we give a fairly simple proof of the Broadhurst-Kreimer conjecture about the values of zig-zag multi-loop two-point diagrams which make a significant contribution to the renormalization group β-function in the 4-dimensional ϕ4 theory.