Regularity of path ideals of gap free graphs

In this paper we study the Castelnuovo–Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the t-path ideals of gap free, claw free and whiskered-$K_4$ free graphs have linear minimal free resolutions for all $t\ge 3$.