Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo–Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is claw free with no induced 4-cycle in its complement. For the last family we show that the second power of the edge ideal has a linear resolution.