The relationships, in many cases equivalences, between lattice distributivity, adjunction and continuity have been studied by many authors, for example [1, 3–8, 12, 13, 15, 17–20, 22, 23]. Very roughly, we refer to the following circle of ideas. Let L be an ordered set, and L a class of subsets of L, and suppose that L has a supremum for each element in L. We might say that L has -sups. The ‘distributivity’ we refer to is that of infs over -sups. The ‘adjunction’ is that given by a left adjoint to the map V: L→L. Now the latter has a left adjoint if and only if it preserves infs, and this means roughly that the -sup of an intersection is an inf of -sups. When one does succeed in identifying the -sup of an intersection as a -sup of infs, one has an instance of distributivity.

The graphs are undirected, without loops or multiple edges. The edge set E(X) of a graph X is a set of certain unordered pairs [x, y] of distinct elements of the vertex set V(X). For x ϵ V(X) we denote by E(x; X) the edges of X incident with x. A (homo)morphism ϕ : X ⟶ Y is a function from V(X) to V(Y) which preserves edges; thus it induces ϕ# : E(X) ⟶ E(Y) by ϕ# [x, x’] = [ϕx, ϕx’].

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The surjectivity of epimorphisms in the category of planar graphs and edge-preserving maps follows from and is implied by the Four Colour Theorem. The argument that establishes the equivalence is not combinatorially complex.

This paper answers affirmatively a question of Pavol Hell [2]: if a graph admits a full n-colouring for every finite n ≧ n0, does it admit an infinité full colouring? (A colouring is full if every pair of distinct colour classes is joined by at least one edge).