Highly connected counterexamples to the permutation Tutte inequality
We disprove Conjecture 7.3 of Csikvári, Around the Merino–Welsh conjecture: improving Jackson's inequality, which asserts that $\widetilde T_H(2,0)\widetilde T_H(0,2)\ge1$ for every bipartite graph $H$ of minimum degree at least two. For each integer $k\ge2$, we construct an explicit sequence of finite simple balanced bipartite graphs whose minimum degree and vertex connectivity are both exactly $k$. The graphs admit automorphisms interchanging their two vertex classes, and both $\widetilde T_H(2,0)$ and $\widetilde T_H(0,2)$ tend to zero along the sequence. Consequently, no fixed…
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