Gallai-Milgram Theorem Research Articles

A unified approach to known and unknown cases of Berge's conjecture

AbstractBerge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection Ck of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the Greene–Kleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai–Milgram Theorem, for k⩾λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai–Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|⩾k. Recently, it was proved (Eur J Combin (2007)) for k=2.There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ−1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.

Paths and Stability Number in Digraphs

The Gallai-Milgram theorem says that the vertex set of any digraph with stability number $k$ can be partitioned into $k$ directed paths. In 1990, Hahn and Jackson conjectured that this theorem is best possible in the following strong sense. For each positive integer $k$, there is a digraph $D$ with stability number $k$ such that deleting the vertices of any $k-1$ directed paths in $D$ leaves a digraph with stability number $k$. In this note, we prove this conjecture.

Lexicographic Products and a Conjecture of Hahn and Jackson

The Gallai-Milgram theorem asserts that the vertex set of any digraph with stability number $k$ can be partitioned into $k$ directed paths. Hahn and Jackson conjectured that, for any positive integer $k$, there exists a digraph with stability number $k$ such that the subdigraph obtained by deleting any $k-1$ directed paths still has stability number $k$. They established the existence of such digraphs for $k=1,2,3$. Here we construct examples for arbitrarily large values of $k$.

Paths partition with prescribed beginnings in digraphs: A Chvátal–Erdős condition approach

A digraph D verifies the Chvátal–Erdős conditions if α ( D ) ⩽ κ ( D ) , where α ( D ) is the stability number of D and κ ( D ) is its vertex-connectivity. Related to the Gallai–Milgram Theorem (see Gallai and Milgram [Verallgemeinerung eines Graphentheorischen Satzes von Redei, Acta Sci. Math. 21 (1960) 181–186]), we raise in this context the following conjecture. For every set of α = α ( D ) vertices { x 1 , … , x α } , there exists a vertex-partition of D into directed paths { P 1 , … , P α } such that P i begins at x i for all i. The case α ( D ) = 2 of the conjecture is proved.

Proof of Berge’s strong path partition conjecture for k=2

Berge’s strong path partition conjecture from 1982 generalizes and extends Dilworth’s theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture is known to be true for all digraphs only for k = 1 (by the Gallai–Milgram theorem) and for k ≥ λ (where λ is the cardinality of the longest path in the graph). The attempts made, so far, to prove the conjecture for other values of k have yielded proofs for acyclic digraphs, but not for general digraphs. In this paper, we prove the conjecture for k = 2 for all digraphs. The proof is constructive and it extends the proof for k = 1 .

Minmax relations for cyclically ordered digraphs

We prove a range of minmax theorems about cycle packing and covering in digraphs whose vertices are cyclically ordered, a notion promoted by Bessy and Thomassé in their beautiful proof of the following conjecture of Gallai: the vertices of a strongly connected digraph can be covered by at most as many cycles as the stability number. The results presented here provide relations between cycle packing and covering and various objects in graphs such as stable sets, their unions, or feedback vertex- and arc-sets. They contain the results of Bessy and Thomassé with simple algorithmic proofs, including polynomial algorithms for weighted variants, classical results on posets extending Greene and Kleitman's theorem (that in turn contains Dilworth's theorem), and a common generalization of these. The most general minmax results concern the maximum number of vertices of a k-chromatic subgraph ( k ∈ N ) —as a consequence, this number is greater than or equal to the minimum of | X | + k | C | , running on subsets of vertices X and families of cycles C covering all vertices not in X, in strongly connected digraphs. This is the “circuit cover” version of a conjecture of Linial (like Gallai's conjecture is the circuit cover version of the Gallai–Milgram theorem, meaning that path partitions are replaced by circuit covers under strong connectivity); we also deduce the circuit cover version of a conjecture of Berge on path partitions; these conjectures remain open, but the proven statements also bound the maximum size of a k-chromatic subgraph, contain Gallai's conjecture, and Bessy and Thomassé's theorems. All presented minmax relations are proved using cyclic orders and a unique elementary argument based on network flows—algorithmically only shortest paths and potentials in conservative digraphs—varying the parameters of the network flow model. In this way antiblocking and blocking relations can be established, leading to a general polyhedral phenomenon—a combination of “integer decomposition, integer rounding” and “dual integrality”—that also contains the matroid partition theorem and Dilworth's theorem. We provide a common reason for all these minmax equalities to hold and some possible other ones which satisfy the same abstract properties.

A short proof of the Chen-Manalastas theorem

Gallai and Milgram (1960) proved that a digraph with stability number α is spanned by α disjoint directed paths. Chen and Manalastas Jr (1983) proved that a strong digraph with stability number at most two is spanned by at most two consistent directed circuits. We slightly simplify the proof of the Gallai-Milgram theorem, while at the same time refining its statement, and use this sharpened version to obtain a relatively short proof of the Chen-Manalastas theorem. We also give a counterexample to a generalization of the Gallai-Milgram theorem conjectured by Hartman (1988).

Coflow polyhedra

A system L of linear inequalities in the variables x is called totally dual integral (TDI) if for every linear function cx such that c is all integers, the dual of the linear program: maximize { cx≔ x satisfies L} has an integer-valued optimum solution or no optimum solution. A linear system L is called box TDI if L together with any inequalities b⩽ x⩽ a is TDI. The main result of this paper is the Coflow Polyhedron Theorem: For any digraph G with node-set V ( G), and for any fixed rational-valued d> = ( d v ≔ v ϵ V ( G)), the following system of inequalities in variables x = ( x v ≔ v ϵ V ( G)) is box TDI. ∀ dicircuit C of G, ∑{x v≔v∈C} ⩽ ∑ {d v≔v∈C}. By Edmonds and Giles' basic theorem on TDI systems, the Coflow Polyhedron Theorem provides many combinatorial min-max relations. In particular, it implies min-max theorems for the maximum weight union of k antichains and the maximum weight union of k chains in a poset. It allows us to prove that a class of graphs which generalize comparability graphs are perfect, and prove a theorem which for acyclic digraphs generalizes the Gallai–Milgram Theorem. The proof of dual integrality of the inequality system described above uses network flow techniques. The proof of primal integrality then follows by the Edmonds-Giles Theorem, or can be proved directly using the concept of projection of a polyhedron.

Gallai-Milgram properties for infinite graphs

We discuss extensions of the Gallai-Milgram theorem to infinite graphs. We define a path to be a directed graph whose transitive closure is a linear ordering. We show that an undirected graph with no infinite independent set is covered by finitely many pairwise disjoint paths; moreover for a given integer k this graph is covered by ⩽k (resp. ⩽k pairwise disjoint) paths if each finite set of vertices is contained in the union of ⩽k (resp. ⩽k pairwise disjoint) paths. Hence an undirected graph with no independent set of size k + 1 is covered by ⩽k pairwise disjoint paths. We prove also that if to each edge (x, y) of a countable path is associated an element θ(x, y) of a finite group, then some edges can be deleted so that the new graph is still a path and the product θ(x0, x1)•θ(x1, x2)•⋯•θ(xn−1, xn) of the elem ents of the group along any finite path (x0, x1,…,xn) of the new graph depends only upon the extremities x0, x>n of the path.

Variations on the Gallai-Milgram theorem

We strengthen the Gallai-Milgram theorem for digraphs with independence number two, and propose various problems and conjectures strengthening the Gallai-Milgram theorem and the Gallai-Roy theorem for strong digraphs.

Extending the Greene-Kleitman theorem to directed graphs

The celebrated Dilworth theorem ( Ann. of Math. 51 (1950), 161–166) on the decomposition of finite posets was extended by Greene and Kleitman ( J. Combin. Theory Ser. A 20 (1976), 41–68). Using the Gallai-Milgram theorem ( Acta Sci. Math. 21 (1960), 181–186) we prove a theorem on acyclic digraphs which contains the Greene-Kleitman theorem. The method of proof is derived from M. Saks' elegant proof ( Adv. in Math. 33 (1979), 207–211) of the Greene-Kleitman theorem.

Sur les arborescences dans un graphe oriente

The main result of this paper is the following theorem: Let G = ( X, E) be a digraph without loops or multiple edges, | X| ⩾3, and h be an integer ⩾1, if G contains a spanning arborescence and if d + G ( x)+ d − G ( x)+ d − G ( y)+ d − G ( y)⩾ 2| X |−2 h−1 for all x, y ϵ X, x ≠ y, non adjacent in G, then G contains a spanning arborescence with ⩽ h terminal vertices. A strengthening of Gallai-Milgram's theorem is also proved.