Before revision:  The constraint density where algorithms break down for each CSP is known as the "algorithmic barrier" and provably corresponds to a phase transition in the geometry of the space of solutions Achlioptas and Coja-Oghlan 2008 . In this paper we aim to shed light on the following question: Can algorithmic success up to the barrier for each CSP be ascribed to some simple structural property of the inputs? We answer this question positively for graph coloring by identifying the property ofgirth-reducibility .
After revision:  The constraint density where algorithms break down for each CSP is known as the "algorithmic barrier" and provably corresponds to a phase transition in the geometry of the space of solutions Achlioptas and Coja-Oghlan 2008 girth-reducibility .