Schubert graphs, symmetric groups and flags of Boolean lattices (Q1312225)

Given a positive integer \(k\), a Schubert graph of index \(k\) is a simple, connected and \(k\)-regular graph which admits a \(k\)-edge-colouring satisfying the following conditions: (1) any 2-coloured path of length 4 whose colours are not consecutive is closed; (2) the colours of any closed 2-coloured path of length 4 are not consecutive; (3) any 2- coloured path of length 6 is closed: (4) given \(i\in\{1,2,\dots,k\}\), any path with exactly one edge coloured \(i\) and without edges coloured \(j\), for \(j>i\), is open. The author proves that a Schubert graph of index \(k\) is unique up to isomorphism and that this graph is isomorphic to the graph representing the flags of a Boolean lattice. Moreover, the concept of Schubert graphs is used for a new proof of a well-known characterization of symmetric groups.