  A convenient integral 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 beta-function in 4-dimensional phi^4 theory.
