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We prove that every graph G has a vertex partition into a cycle and an anticycle (a cycle in the complement of G). Emptyset, singletons and edges are considered as cycles. This problem was posed by Lehel and shown to be true for very large graphs by Luczak, Rödl and Szemerédi [7], and more recently for large graphs by Allen [1]. Many questions deal with the(More)
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. 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(More)
Given a graph G = (V, E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (V × V) \ E such that the graph H = (V, E ∪ F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research [16, 24]. First announced by Kaplan,(More)
A tournament T = (V, A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is(More)
A graph G = (V, E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u, v) ∈ E iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification.(More)
A k-digraph is a digraph in which every vertex has outdegree at most k. A (k ∨ l)-digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ(k) (resp. Φ ∨ (k, l)) of the arc-chromatic number over the k-digraphs (resp. (k ∨ l)-digraphs). El-Sahili [3] showed that Φ ∨(More)
The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to(More)
Answering a question of Adrian Bondy [4], we prove that every strong digraph has a spanning strong subgraph with at most n + 2α − 2 arcs, where α is the size of a maximum stable set of D. Such a spanning subgraph can be found in polynomial time. An infinite family of oriented graphs for which this bound is sharp was given by Odile Favaron [3]. A direct(More)