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 (J₆) reg (J₇_₁) + reg (J₇_₂) -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 (J₆) 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 (J₆), namely that reg (J₆) ht (J₆) +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 J₆ 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 (J₆) given by the size of a maximal clique disjoint set of edges.