A-computable graphs

We consider locally finite graphs with vertex set N. A graph G is computable if the edge set is computable and highly computable if the neighborhood function NG (which given v outputs all of its adjacent vertices) is computable. Let χ(G) be the chromatic number of G and χ(G) be the computable chromatic number of G. Bean showed there is a computable graph G with χ(G) = 3 and χ(G) =∞, but if G is highly computable then χ(G) ≤ 2χ(G). In a computable graph the neighborhood function is ∆2. In highly computable graphs it is computable. It is natural to ask what happens between these extremes. A computable graph G is A-computable if NG ≤T A. Gasarch and Lee showed that if A is c.e. and not computable then there exists an A-computable graph G such that χ(G) = 2 but χ(G) =∞. Hence for A noncomputable and c.e., A-computable graphs behave more like computable graphs than highly computable graphs. We prove analogous results for Euler paths and domatic partitions. Gasarch and Lee left open what happens for other ∆2 sets A. We show that there exists an ∅ <T A <T ∅′ such that every A-computable graph G with χ(G) <∞ has χ(G) <∞. Finally, we classify all such A.