Spanning a strong digraph by α circuits: A proof of Gallai’s conjecture

In 1963, Tibor Gallai [9] asked whether every strongly connected directed graph D is spanned by α directed circuits, where α is the stability of D. We give a proof of this conjecture. 1 Coherent cyclic orders. In this paper, circuits of length two are allowed. Since loops and multiple arcs play no role in this topic, we will simply assume that our digraphs are loopless and simple. A directed graph (digraph) is strongly connected, or simply strong, if for all vertices x, y, there exists a directed path from x to y. A stable set of a directed graph D is a subset of vertices which are not pairwise joined by arcs. The stability of D, denoted by α(D), is the number of vertices of a maximum stable set of D. It is well-known, by the Gallai-Milgram theorem [10] (see also [1] p. 234 and [3] p. 44), that D admits a vertex-partition into α(D) disjoint paths. We shall use in our proof a particular case of this result, known as Dilworth’s theorem [8]: A partial order P admits a vertex-partition into α(P ) chains (linear orders). Here α(P ) is the size of a maximal antichain. In [9], Gallai raised the problem, when D is strongly connected, of spanning D by a union of circuits. Precisely, he made the following conjecture (also formulated in [1] p. 330, [2] and [3] p. 45): Conjecture 1 Every strong digraph with stability α is spanned by the union of α circuits. The case α = 1 is Camion’s theorem [6]: Every strong tournament has a hamilton circuit. The case α = 2 is a corollary of a result of Chen and Manalastas [7] (see also Bondy [4]): Every strong digraph with stability two is spanned by two circuits intersecting each other on a (possibly empty) path. In [11] was proved the case α = 3. In the next section of this paper, we will give a proof of Gallai’s conjecture for every α. Let D be a strong digraph on vertex set V . An enumeration E = v1, . . . , vn of V is elementary equivalent to E′ if one the following holds: E′ = vn, v1, . . . , vn−1, or E′ = v2, v1, v3, . . . , vn if neither v1v2 nor v2v1 is an arc of D. Two enumerations E,E′ of V are equivalent if there is a sequence E = E1, . . . , Ek = E′ such that Ei and Ei+1 are elementary equivalent, for i = 1, . . . , k − 1. The classes of this equivalence relation are called the cyclic orders of D. Roughly speaking, a cyclic order is a class of enumerations of the vertices on the integers modulo n, where one stay in the class while switching consecutive vertices which are not joined by an arc. We fix an enumeration E = v1, . . . , vn of V , the following definitions are understood with respect to E. An arc vivj of D is a forward arc if i < j, otherwise it is a backward arc. A directed path of D is a forward path if it only contains forward arcs. The index of a directed circuit C of D is the number of backward arcs of C, we denote it by iE(C).