LF-Covers and Binomial Edge Ideals of König Type

We give a combinatorial characterisation of connected graphs whose binomial edge ideals are of König type, developed independently to the similar characterisation given by LaClair in arXiv:2304.13299, and exhibit some classes of graphs satisfying our criteria. For any connected Hamiltonian graph $G$ on $n$ vertices, we compute an explicit root of $H_{J(G)}^{n-1}(R)$ as an $F$-finite $F$-module.