LF-Covers and Binomial Edge Ideals of K\'{o}nig Type

We give a combinatorial characterisation of connected graphs whose binomial edge ideals are of K\"{o}nig 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.
Comment: 19 pages, v3 substantially reorganises v2, several proofs have been simplified or rewritten for clarity, and small errors have been corrected (whilst the numbering has changed, no new results are introduced)