A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle ofâ€¦ (More)

The tree-width of a graph G is the minimum k such that G may be decomposed into a " tree-structure " of pieces each with at most k + 1 vertices. We prove that this equals the maximum k such thatâ€¦ (More)

In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t+1 vertices is t-colourable. When t â‰¤ 3 this is easy, and when t = 4, Wagnerâ€™s theorem ofâ€¦ (More)

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give anotherâ€¦ (More)

* Partially supported by NSF under Grant No. DMS-9701598. â€ Research partially supported by NSF under Grant No. DMS-9401981 and by DIMACS Center, Rutgers University, New Brunswick, NJ 08903. â€¡â€¦ (More)

Let G be an n-vertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given h-vertex graph H. We prove that there is a set X of no more thanâ€¦ (More)

Let G be an n-vertex graph with no minor isomorphic to an hvertex complete graph. We prove that the vertices of G can be partitioned into three sets A, B , C such that no edge joins a vertex in Aâ€¦ (More)