On the structure of 5- and 6-chromatic abstract graphs.

The (abstract) graphs considered in this paper are undirected and have no loops and no multiple edges. An edge by itself without its end-vertices is not considered to be a graph, a set of vertices without edges is a graph. An edge is said to join its two end-vertices. The valency of a vertex is the number of vertices joined to it by edges. A <κ>, where κ is a natural number, is a graph with κ vertices in which each pair of distinct vertices are joined by an edge; a <κ—> is a <κ> with just one edge missing. A path is a graph with vertices a1? . . ., αμ and edges (α1? α2), (α2, α8), . . ., (α μ_ι, αμ), where μ Ξ> 2 and α1? . . ., αμ are all distinct. αι and αμ are called the end-vertices of the path, the others are called its intermediate vertices. The length of the path is μ — l, and a path is even or odd according to whether its length is even or odd. A circuit is a graph with vertices a l 7 . . ., av and edges (%, α2), (α2, α3), . . ., (αν_1? a„), (av, ax), where v^>3 and # ! , . . . , # „ are all distinct. The length of the circuit is v, and a circuit is even or odd according to whether its length is even or odd. If A and B are two non-empty disjoint graphs then any path which has one end-vertex in A and the other end-vertex in B and has no vertex except its two end-vertices in common with A w B is called an (-4) (B)path or (B) (A)-path, in particular a path with end-vertices a and b is an (a) (fr)-path and (b) (a)-path. A set of vertices is called independent if no two are joined by an edge.