A famous theorem of Kalai and Meshulam is that $\mathrm{ reg}(I + J) \leq \mathrm{ reg}(I) + \mathrm{ reg}(J) -1$ 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 $\mathrm{ reg}(J_{ G}) \leq \mathrm{ reg}(J_{ H_{ 1}}) + \mathrm{ reg}(J_{ H_{ 2}}) -1$ whenever $G$, $H_{ 1}$, and $H_{ 2}$ are graphs satisfying $E(G) = E(H_{ 1}) \cup E(H_{ 2})$ and $J_{ \ast}$ is the associated binomial edge ideal. We prove a special case of this conjecture which strengthens the celebrated theorem of Malayeri--Madani--Kiani that $\mathrm{ reg}(J_{ G})$ 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 $\mathrm{ reg}(J_{ G})$, namely that $\mathrm{ reg}(J_{ G}) \leq \mathrm{ ht}(J_{ G}) +1$. Our upper bound gives an analogue of the well-known result that $\mathrm{ reg}(I(G)) \leq \mathrm{ ht}(I(G)) +1$ 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 $J_{ G}$ 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 $\mathrm{ reg}(J_{ G})$ given by the size of a maximal clique disjoint set of edges.
Categories: math.AC