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

A famous theorem of Kalai and Meshulam is that $\mathrm{r}\mathrm{e}\mathrm{g}(I+J)\le \mathrm{r}\mathrm{e}\mathrm{g}(I)+\mathrm{r}\mathrm{e}\mathrm{g}(J)-1$ 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 $\mathrm{r}\mathrm{e}\mathrm{g}(J_G)\le \mathrm{r}\mathrm{e}\mathrm{g}(J_{H_1})+\mathrm{r}\mathrm{e}\mathrm{g}(J_{H_2})-1$ 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 $\mathrm{r}\mathrm{e}\mathrm{g}(J_G)$ 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 $\mathrm{r}\mathrm{e}\mathrm{g}(J_G)$, namely that $\mathrm{r}\mathrm{e}\mathrm{g}(J_G)\le \mathrm{h}\mathrm{t}(J_G)+1$. Our upper bound gives an analogue of the well-known result that $\mathrm{r}\mathrm{e}\mathrm{g}(I(G))\le \mathrm{h}\mathrm{t}(I(G))+1$ 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 $\mathrm{r}\mathrm{e}\mathrm{g}(J_G)$ given by the size of a maximal clique disjoint set of edges.