We went over an example of a bipartite graph (a crown graph) for which a greedy coloring with a bad vertex ordering can produce a non-optimal coloring. But the bad ordering that we used for this graph was not the lexicographic breadth first search ordering (which works well for interval graphs). Prove that, when a breadth first ordering is used (regardless of whether it is the lexicographic breadth first search ordering), the greedy coloring algorithm with this ordering will optimally color every bipartite graph.