Linkedness and Ordered Cycles in Digraphs

Given a digraph D, let δ0(D) := min{δ+(D), δ−(D)} be the minimum semi-degree of D. We show that every sufficiently large digraph D with δ0(D)≥n/2 + l −1 is l-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis [17]. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e., that for every sequence s1,. . ., sk of distinct vertices of D there is a directed cycle which encounters s1,. . ., sk in this order. This result will be used in [16].

[1]  Alexandr V. Kostochka,et al.  An extremal problem for H‐linked graphs , 2005, J. Graph Theory.

[2]  Michael S. Jacobson,et al.  Linear forests and ordered cycles , 2004, Discuss. Math. Graph Theory.

[3]  Daniela Kühn,et al.  k-Ordered Hamilton cycles in digraphs , 2007, J. Comb. Theory, Ser. B.

[4]  Gregory Gutin,et al.  Some Parameterized Problems On Digraphs , 2008, Comput. J..

[5]  Ken-ichi Kawarabayashi,et al.  On Sufficient Degree Conditions for a Graph to be $k$-linked , 2006, Combinatorics, Probability and Computing.

[6]  H. Jung.  Eine Verallgemeinerung desn-fachen Zusammenhangs für Graphen , 1970 .

[7]  Colton Magnant,et al.  Pan-H-Linked Graphs , 2010, Graphs Comb..

[8]  Yannis Manoussakis,et al.  k-Linked and k-cyclic digraphs , 1990, J. Comb. Theory, Ser. B.

[9]  Paul Wollan,et al.  An improved linear edge bound for graph linkages , 2005, Eur. J. Comb..

[10]  Hong Wang,et al.  Vertex-Disjoint Cycles Containing Specified Edges , 2000, Graphs Comb..

[11]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[12]  Carsten Thomassen,et al.  Highly connected non-2-linked digraphs , 1991, Comb..

[13]  Gábor N. Sárközy,et al.  On k-ordered Hamiltonian graphs , 1999, J. Graph Theory.

[14]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[15]  Alexandr V. Kostochka,et al.  Minimum degree conditions for H-linked graphs , 2005, Discret. Appl. Math..

[16]  M. Meyniel.  Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe oriente , 1973 .

[17]  Thor Whalen,et al.  Graphs and Combinatorics , 2007 .

[18]  Marie-Claude Heydemann,et al.  About some cyclic properties in digraphs , 1985, J. Comb. Theory, Ser. B.

[19]  Alexandr V. Kostochka,et al.  On Minimum Degree Implying That a Graph is H-Linked , 2006, SIAM J. Discret. Math..

[20]  Béla Bollobás,et al.  Highly linked graphs , 1996, Comb..

[21]  Michael Ferrara,et al.  On H-Linked Graphs , 2006, Graphs Comb..