Regularity of binomial edge ideals of chordal graphs

In this paper we prove the conjectured upper bound for Castelnuovo–Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques of G, denoted by c(G). Moreover, we classify all chordal graphs G for which $$\mathcal {L}(G)=c(G)$$, where $$\mathcal {L}(G)$$ is the sum of the lengths of longest induced paths of connected components of G. We call such graphs strong interval graphs. We show that the regularity of a strong interval graph G coincides with $$\mathcal {L}(G)$$ as well as c(G).