Origami edge-paths in the curve graph

An origami (or flat structure) on a closed oriented surface, Sg, of genus g≥2 is obtained from finite collection unit Euclidean squares by gluing each right edge to left one and top bottom one. The main objects study in this note are pairs curves—filling simple curves, (α,β), Sg such that their minimal intersection equal algebraic intersection—they coherent. pair curves naturally associated with an Sg. Our result establishes for any there exists edge-path, sequence α=α0,α1,α2,⋯,αn=β, that: αi intersects αi+1 exactly once; (αi,αj) coherent; thus, filling pair, (αi,αj), also origami. With existence established, we offer shortest edge-paths as area investigation.