A new upper bound for the regularity of gap-free graphs

In this article, we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of the their minimal triangulation. Let $H_U=G\cup F_U$ be a minimal triangulation of a gap-free graph $G$, for some maximal independent set $U$ in $G$. Let $\mathcal{C}_U$ be the $3$-uniform clutter of all $3$-paths in $H_U$ which consists of one edge coming from $F_U$ and another edge coming from $G$. Then we show that $\displaystyle \reg(I(G))\leq \reg(I(\C_U))$. As a consequence, we give a general upper bound for the regularity of gap-free graphs. Furthermore, if $\mathcal{H}$ is the $3$-uniform clutter consists of the $3$-cliques in $G$ or in $F_U$, and the $3$-paths in $G$ which are not $3$-cliques in $H_U$, then $\reg(I(G))\leq 3$, provided $\mathcal{H}$ is chordal. This answers partially a question raised by H\'a, \cite[Problem $6.3$]{h14} and by Banerjee, Beyarslan and H\'a, \cite[Problem $7.1$]{bbh19}.