A New Conjecture and Upper Bound on the Castelnuovo--Mumford Regularity of Binomial Edge Ideals

A famous theorem of Kalai and Meshulam is that reg(I + J) ≤ reg(I) + reg(J) -1 for any squarefree monomial ideals I and J. This result was subsequently extended by Herzog to the case where I and J are any monomial ideals. In this paper we conjecture that the Castelnuovo--Mumford regularity is subadditive on binomial edge ideals. Specifically, we propose that reg(JG) ≤ reg(J_H₁) + reg(J_H₂) -1 whenever G, H₁, and H₂ are graphs satisfying E(G) = E(H₁) ∪ E(H₂) and J_ is the associated binomial edge ideal. We prove a special case of this conjecture which strengthens the celebrated theorem of Malayeri--Madani--Kiani that reg(JG) is bounded above by the minimal number of maximal cliques covering the edges of the graph G. From this special case we obtain a new upper bound for reg(JG), namely that reg(JG) ≤ ht(JG) +1. Our upper bound gives an analogue of the well-known result that reg(I(G)) ≤ ht(I(G)) +1 where $I(G)$ is the edge ideal of the graph G. We additionally prove that this conjecture holds for graphs admitting a combinatorial description for its Castelnuovo--Mumford regularity, that is for closed graphs, bipartite graphs with JG Cohen--Macaulay, and block graphs. Finally, we give examples to show that our new upper bound is incomparable with Malayeri--Madani--Kiani's upper bound for reg(JG) given by the size of a maximal clique disjoint set of edges.