Hamiltonian Cycles Avoiding a Spanning Forest or a Spanning Cycle Subgraph in Tournaments - Di, Yaoxiang; Li, Ruijuan; Zhang, Xinhong (Graphs and combinatorics, 2025)
Bang-Jensen, Gutin and Yeo [Combin. Probab. Comput. 6(3) (1997) 255–261] investigated hamiltonian cycles avoiding the union of disjoint cliques in tournaments: for a k -strong tournament T = ( V , A ) on n vertices, a partition X 1 , X 2 , … , X p of V with | X 1 | ≤ | X 2 | ≤ ⋯ ≤ | X p | , and a digraph D obtained from T by deleting all arcs which have both head and tail in the same X i (i.e., D = T - ∪ i = 1 p A ( T [ X i ] ) ), if | X p | ≤ n / 2 and k ≥ | X p | + ∑ i = 1 p - 1 | X i | / 2 , then D is hamiltonian. They showed the bound on k is the best possible and raised the problem: which sets B of edges of the complete graph K n have the property that every k -strong orientation of K n induces a hamiltonian digraph on K n - B ? The above result provides the sharp bound of k when B is the union of cliques. In particular, they asked what are sharp bounds for k when B is a spanning forest (or a spanning cycle subgraph) of K n , consisting of p disjoint paths, or p disjoint stars (or p disjoint cycles) containing r 1 , … , r p vertices, respectively. In this paper, we give the bounds for k on the above problems and prove these bounds are almost best possible, when each component of the spanning forest (or spanning cycle subgraph) has at most 3 vertices.