II. Stable sets and colourings

We have seen that in any graph G = (V,E), a maximum-size matching can be found in polynomial time. This means that α(L(G)) can be found in polynomial time, where L(G) is the line graph of G.2 On the other hand, it is NP-complete to find a maximum-size stable set in a graph. That is, determining α(G) is NP-complete. Since α(G) = |V |−τ(G) and α(G) = ω(G), also determining the vertex cover number τ(G) and the clique number ω(G) are NP-complete problems. Moreover, determining χ(G) is NP-complete. It is even NP-complete to decide if a graph is 3-colourable. Note that one can decide in polynomial time if a graph G is 2-colourable, as bipartiteness can be checked in polynomial time. These NP-completeness results imply that if NP 6=co-NP, then one may not expect a min-max relation characterizing the stable set number α(G), the clique number ω(G), or the colouring number χ(G) of a graph G.