Nontraceable detour graphs

The detour order (of a vertex vv) of a graph G   is the order of a longest path (beginning at vv). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order. Homogeneously traceable graphs are therefore detour graphs with detour deficiency zero. In this paper, we give a number of constructions for detour graphs of all orders greater than 17 and all detour deficiencies greater than zero. These constructions are used to give examples of nontraceable detour graphs with chromatic number k  , k⩾2k⩾2, and girths up to 7. Moreover we show that, for all positive integers l⩾1l⩾1 and k⩾3k⩾3, there are nontraceable detour graphs with chromatic number k and detour deficiency l.