Puzzling and apuzzling graphs

Let GG be a graph with chromatic number χ(G)χ(G) and consider a partition PP of GG into connected subgraphs. PP is a puzzle on GG if there is a unique vertex coloring of GG using 1, 2, …, χ(G)χ(G) such that the sums of the numbers assigned to the partition pieces are all the same. PP is an apuzzle if there is a unique vertex coloring such that the sums are all different.We investigate the concept of puzzling and apuzzling graphs, detailing classes of graphs that are puzzling, apuzzling and neither.