Transition Restricted Gray Codes

A Gray code is a Hamilton path H on the n-cube, Q n. By labeling each edge of Q n with the dimension that changes between its incident vertices, a Gray code can be thought of as a sequence H = t 1 ; t 2 ; : : : ; t N-1 (with N = 2 n and each t i satisfying 1 t i n). The sequence H deenes an (undirected) graph of transitions, G H , whose vertex set is f1; 2; : : : ; ng and whose edge set E(G H) = ft i ; t i+1 ] j 1 i N-1g. A G-code is a Hamilton path H whose graph of transitions is a subgraph of G; if H is a Hamilton cycle then it is a cyclic G-code. The classic binary reeected Gray code is a cyclic K 1;n-code. We prove that every tree T of diameter 4 has a T-code, and that no tree T of diameter 3 has a T-code.