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spaced-out the long paragraph of text and equations. Corrected one typo. Added clarifying statement that (N+1,N+1) is the midpoint of this lattice grid
It doesn't look to me like that fact was used in the IOI problem ... from what I understood, their example 3 simply says that there is no way to have 3 simple paths between two vertices in a graph with $|V|=2$. Is your fact about your defined $f(u,v)$ verified?
can you elaborate on 'chordless cycles' ? Most definitions refer to a chord of a cycle $v_1,v_2,v_3,v_4,...,v_n,v_1$ to mean an edge from one $v_i$ to another $v_j$ with $|i-j| \neq 1$ (mod n). But I think in your context, you would be counting a path to qualify as a chord. Does the path necessarily have to be a subdivision of a single edge? Can you maybe illustrate your V and P for a graph like {ab,bc,cd,de,ef,fa, gf,gb,he,hc} ?
