Effective matchmaking and 𝑘-chromatic graphs

Abstract

In an earlier paper we showed that there is a recursive society, in which each person knows exactly two other people, whose marriage problem is solvable but not recursively solvable. We generalize this result, using a different construction, to the case where each person knows exactly k k other people. From this we deduce that for each k ≧ 2 k \geqq 2 there is a recursive 2 ( k − 1 ) 2(k - 1) -regular graph, whose chromatic number is k k but which is not recursively k k chromatic.