Infinite Counterexamples to Written on the Wall II Conjecture 194

For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods. Written on the Wall II Conjecture 194 asserts that every connected graph satisfying alpha(G) <= 1 + l_avg(G) has a Hamiltonian path. We give a three-parameter family of counterexamples. A one-parameter subfamily satisfies the proposed inequality with equality: for every integer t >= 3 it has 2t^2 vertices, independence number t + 1, and l_avg(G) = t, but has t pendant vertices and hence no Hamiltonian path. The first member of this subfamily has 18 vertices; that concrete case has also been verified in Lean 4.