A note on some embedding problems for oriented graphs

We conjecture that every oriented graph G on n vertices with δ(G), δ−(G) ≥ 5n/12 contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered. 1.1. Powers of Hamilton cycles. One of the most studied problems in graph theory concerns finding sufficient conditions that ensure a graph contains a Hamilton cycle. Dirac [4] showed that any graph G on n ≥ 3 vertices has a Hamilton cycle provided that it has minimum degree δ(G) at least n/2. For a digraph G it is natural to consider its minimum semidegree δ0(G), which is the minimum of its minimum outdegree δ+(G) and its minimum indegree δ(G). (The digraphs we consider do not have loops and we allow at most one edge in each direction between any pair of vertices.) Ghouila-Houri [6] proved that every digraph G on n ≥ 2 vertices with δ0(G) ≥ n/2 is Hamiltonian. An important subclass of digraphs is the class of oriented graphs: these are the digraphs which do not contain any 2-cycles. Keevash, Kühn and Osthus [9] showed that any sufficiently large oriented graph G on n vertices with δ0(G) ≥ (3n− 4)/8 is Hamiltonian, thereby proving a conjecture of Häggkvist [7]. For a detailed account of other such results concerning Hamilton cycles in directed and oriented graphs see [13]. A generalisation of the notion of a Hamilton cycle is that of the rth power of a Hamilton cycle. Indeed, the rth power of a Hamilton cycle C is obtained from C by adding an edge between every pair of vertices of distance at most r on C. Seymour [18] conjectured the following strengthening of Dirac’s theorem. Conjecture 1 (Seymour [18]). Let G be a graph on n vertices. If δ(G) ≥ r r+1n then G contains the rth power of a Hamilton cycle. Pósa (see [5]) had earlier proposed the conjecture in the case of the square of a Hamilton cycle (that is, when r = 2). Komlós, Sárközy and Szemerédi [11] proved Conjecture 1 for sufficiently large graphs. The notion of the rth power of a Hamilton cycle also makes sense in the digraph setting: In this case the rth power of a Hamilton cycle C is the digraph obtained from C by adding a directed edge from x to y if there is a path of length at most r from x to y on C. Bollobás and Häggkvist [1] proved that given any ε > 0 and any r ∈ N, all sufficiently large tournaments T on n vertices with δ0(T ) ≥ (1/4 + ε)n contain the rth power of a Hamilton cycle. Date: November 19, 2010. The author was supported by the EPSRC, grant no. EP/F008406/1.