A Note on Perfectly orderable Graphs

A natural way to colour the vertices of a graph is: (i) to impose a linear order < on the vertices, and (ii) to scan the vertices in this order, assigning to each vertex c(,j) the smallest positive integer assigned to no neighbour v(k) of o(j) with z>(k) < t:(,j). This heuristic algorithm is called the greedy colouring algorithm, or the sequential colouring algorithm. One may ask the following question: For which ordered graphs does the sequential colouring algorithm deliver an optimal colouring? This question motivated Chvatal [3] to define a “perfect order”: an order < is [email protected] if for each induced subgraph (H, < ) of (G, < ), the sequential colouring algorithm produces an optimal colouring. Chvatal proved that an order < is perfect if and only if the pair (G, < ) does not contain, as induced subgraph, the chordless path on four vertices tl, h, c, d with LI < b, d < c (this ordered subgraph is called an obstruction). A graph is pe+ctly orderable if it admits a perfect order. There is a somewhat surprising connection (pointed out first by Chvatal in [4]) between perfectly orderable graphs and a well-known theorem in mathematical programming which we are about to explain. A bipartite graph is chordal if it contains