Long Cycles in Digraphs

Introduction By Ghouila-Houri's theorem [10], every strong digraph of order n and with minimum degree at least n is Hamiltonian. Extensions of this theorem can be found in [11, 13, 16, 18]. Nash-Williams [15] raised the problem of describing all the extreme digraphs for Ghouila-Houri's theorem, i.e. the strong non-Hamiltonian digraphs of order n and minimum degree n — 1. Bondy [4] specialized this problem by stating the conjecture that the digraphs Z>5 and Z)7 of Fig. 1 (where an undirected edge represents two directed edges of opposite directions) are the only &-diregular non-Hamiltonian digraphs of order n = 2k+l. He also made the stronger conjectures that D5 and D7 are the only strong nonHamiltonian (n— 1)-regular digraphs of order n and that D5 and D7 are the only &-diregular digraphs of order n = 2k +1 that cannot be decomposed into Hamiltonian cycles. The first of these stronger conjectures includes Camion's theorem that every strong tournament is Hamiltonian, and the second includes Kelly's conjecture on Hamiltonian decomposition of diregular tournaments.