Block symmetries in graph coloring reconfiguration systems

Abstract

In this paper, we study reconfiguration systems for proper graph colorings modified by single-vertex recolorings. Applying a general construction of Ghrist and Peterson [15] to the graph coloring problem, we consider cubical complexes in which cubes represent recolorings of independent sets of vertices. For any choice of base graph and number of colors, the 0-skeleton of the associated coloring state complex is the set of proper colorings, and the 1-skeleton is the transition graph.Appealing to symmetries induced by color permutation, we give two main results regarding the organization of the biconnected components (blocks) of coloring transition graphs and the colorings within those blocks. First, the 1-skeleton of any connected coloring state complex has a unique block whose vertex set is the union of entire classes of isomorphic colorings. We call this block the nucleus. Second, the nucleus of the state complex of any non-empty graph contains a non-trivial cycle. This implies that although the state complex for any reconfiguration system is locally CAT(0) [15], only the very simplest coloring state complexes are globally CAT(0).