On the chromatic number of cube-like graphs

Abstract A cube-like graph is a graph whose vertices are all 2 n subsets of a set E of cardinality n , in which two vertices are adjacent if their symmetric difference is a member of a given specified collection of subsets of E . Many authors were interested in the chromatic number of such graphs and thought it was always a power of 2. Although this conjecture is false (we show a cube-like graph of chromatic number 7), we prove that there is no cube-like graph with chromatic number 3.