Color-critical Graphs and Hereditary Hypergraphs

A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary hypergraph, generalizing the chromatic number. We then show examples of applying the results to new problems and indicate the way to algorithms and refined complexity results for all these examples at the same time.

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1 Introduction

Graphs and digraphs are without loops or parallel edges. Given a hypergraph , where is the power-set of we will call the elements of vertices, and those of hyperedges, . A cover is a family such that . We suppose that is a cover. The minimum number of hyperedges in a cover is denoted . The hereditary closure of is where , and is hereditary, if .

In this paper we study hereditary hypergraphs (sometimes also called independence systems, or a simplicial complexes in the literature). Hyperedges of cardinality will be called singletons, and hyperedges of cardinality are called edges. Deleting a vertex of the hypergraph results in the hypergraph . For hereditary hypergraphs this is the same as deleting from all hyperedges. Like for coloring, for hereditary hypergraphs a minimum cover can be supposed to be a partition of , and we will suppose this! Indeed, a vertex contained in several hyperedges can be deleted from one of these hyperedges. This assumption is of primary importance, since the edges and the singletons play a major role in such partitions.

If is a hereditary hypergraph, we have ; if the first inequality is satisfied with equality for all , we say that is critical.

Given a hypergraph , denote by the set of its edges, . The components of are defined as the those of . These form a partition of , and correspond to the usual hypergraph components: is connected if is connected. Abusing terminology, the vertex-set of a component will also be called component. The maximum size of a matching in a graph is denoted by . We prove Gallai’s ingenious, simple lemma [5] for self-containedness:

Lemma.

If is a connected graph, and for all , then .

Proof : Suppose for a contradiction that is a maximum matching and are not covered by . Let the distance of and be minimum among all maximum matchings and two of their uncovered vertices. Let be a shortest path between and . Clearly, , otherwise the edge could be added to , contradicting the maximality of .

Let be the neighbor of on , and a maximum matching of . Each connected component of the symmetric difference of and is the disjoint union of even paths and even circuits alternating between and . If and are not in the same component of , then interchanging the edges of in the component of , neither nor is covered by a matching edge, leading to the same contradiction as before.

On the other hand, if they are in the same component, the same interchange of the edges leads to a maximum matching that leaves and uncovered, contradicting the minimum choice of the distance between and .

Gallai’s theorem on color-critical graphs [6] is a beautiful statement but its original proof was rather complicated, essentially more difficult than the above lemma on factor-critical graphs [5]. Stehlík [10] gave a simpler proof. We show here that the generalization to hereditary hypergraphs can be shortly reduced to Gallai’s Lemma (Section 2), in addition giving rise to a wide range of known and new examples (Section 3)111The Theorem below and its proof have been included in a more complex framework of an unpublished manuscript [9]. Several occurrences of old and recent, direct special cases of hereditary hypergraphs make it useful to provide an exclusive, short presentation of this general theorem, with some examples of hereditary hypergraphs of interest., to algorithms, and clarifications of their complexity issues.

2 Theorem and Proof

Theorem.

In a connected, hereditary, critical hypergraph . Furthermore, either the inequality is strict and there is a minimum cover without singleton, or the equality holds, and there are minimum covers with only edges and exactly one singleton that can be any vertex.

Proof : For the statement is obvious, so suppose is critical, . Then for all there exists a minimum cover containing : indeed, adding to a minimum cover of , we get a minimum cover of . Consider a minimum cover of (partitioning ), maximizing , where is the set of its non-singleton elements. Clearly, .

Claim 1. For all and each component of

Indeed, if and are the number of hyperedges in this component of and respectively, then for if say , then

in the minimum cover consisting of and singletons,

replace the hyperedges in , that is, hyperedges of and singletons,

by the hyperedges of in , and singletons, leading to a cover of size

 ρ+kv−ku+|C∖Cv|−|C∖Cu|<ρ,

a contradiction. But then the proven equality implies that the same replacement – in either directions – of the hyperedges lead to a minimum cover, increasing the size of if say , and this contradiction with the choice of proves the claim.

Claim 2. If each minimum cover of contains a singleton, then for all .

Let be arbitrary, and let us prove that for the hyperedge , . Since , by Claim 1, the component of containing also contains a vertex . Let be a shortest path between and in (in the connected component ). Let be the vertices of , in fact , in this order, necessarily alternating between subsets of hyperedges in and subsets of hyperedges in . We prove by induction on the assertion that the latter subsets (of hyperedges in ) are in fact in :

Note first that , because if it was a subset of an edge , , then replacing and in the minimum cover by , , we get a minimum cover, where the hyperedges of size at least two cover contradicting the definition of , provided . If , can occur in Claim 2 choosing any ; exists, since because of the condition of Claim 2.

This proves the assertion for . Let , and ; is a minimum cover of maximizing the union of non-singletons, and . Now the induction hypothesis finishes the proof of the assertion and of Claim 2.

To finish the proof note first that a minimum cover without singleton implies and we are done. Otherwise, Claim 2 can be applied, and follows for all . This formula also shows that a larger matching would provide a smaller cover. So is a maximum matching of and does not cover , so for all . The connectivity of means by definition that is connected, so the conditions of Gallai’s Lemma are satisfied for : is factor-critical, and with a perfect matching of provide a cover of size .

Let us restate the inequality of the Theorem so that it directly contains the formulation of [6]:

Corollary 2.1.

A hereditary hypergraph with is either not critical, or not connected.

3 Examples, Algorithms and Conclusions

In this section we show some examples applying the results to particular hypergraphs. Any hereditary hypergraph is an example, so we cannot seek for completeness, but we try to show how the specialization works. An important surprise is that it turned out that the role of larger hyperedges is secondary, plays the main role: the covers appearing in the Theorem consist only of edges; in the corollaries the components and connectivity depend only on .

Corollary 3.1.

Let be a hereditary hypergraph with . Then either there exists so that , or is not connected, that is, there exists a partition of so that for all , .

3.1 Hereditary hypergraphs from graphs

Let be either an undirected graph or a digraph, the context always determines the current meaning, and we define hereditary hypergraphs on . The more deeply the hypergraphs are related to , the more interesting the results. Fix a (not necessarily finite) set of (di)graphs and for each (di)graph , let , where

 F:={U⊆V(G):U induces in G a graph without any induced % subgraph in G}.

When are the Theorem or its corollaries meaningful or even interesting for ?

If neither of the -vertex graphs are in , hypergraphs are connected for every graph , and our Theorem and its corollaries are trivial. On vertices there are two undirected graphs: one without, and one with an edge between the two vertices. If the only graph of on two vertices is the edge-less graph, consists of cliques of ; if it is the edge on two vertices, consists of stable sets of . In turn, according to Corollary 3.1, in the former case the disconnectivity of means the disconnectivity of , and in the latter case it means the disconnectivity of the complement of . In these cases, the Theorem specializes to Gallai’s theorem.

It is easy to see that in these cases the only possibility to add to more graphs is to add a clique (or stable set) of given size. Then for some , is the family of cliques or stable sets of size at most on . For the Theorem applies without change and it is then about coloring with at most vertices of each color.

3.2 Hereditary hypergraphs from digraphs

Similarly, for digraphs one of the subgraphs on two vertices has to be excluded: there are now three digraphs on two vertices: with or without an arc as in the undirected case, or with an arc in both directions (-cycle). If contains only the latter, we also do not get anything new: keeping only arcs in both directions as an undirected edge we reduce the problem to Gallai’s colorings in undirected graphs. However, if there are some other graphs in we have three interesting special cases: cliques, stable sets (Gallai), a third case we discuss below as also cases from multigraphs.

Corollary 3.2.

Let be a graph, a set of graphs, , and . Then either there exists so that , or is not connected, that is, there exists a partition of so that for all , induces a graph in .

As argued before the corollary, the interesting cases are when the unique graph on two vertices of is an edge, a non-edge or a -cycle, and in the last case there are many possibilities to exclude further induced subgraphs. For instance we can include in -cycles and all graphs on vertices having -cycles. Actually an arbitrary subset of graphs having directed cycles, or the set of all such graphs can be contained in , and will not make any change in the relevant critical graphs (as compared to including only - and -cycles, no larger hyeredge plays a role). Corollary 3.2 holds, and partitioning into hyperedges of means then partitioning into vertex-sets that induce acyclic digraphs: this is “digraph coloring”, for which Corollary 3.2 was asked in [2]. (The Theorem has then already been proved, see footnote 1. Stehlík [11] missed its specialization to acyclic induced subgraphs, and answered [2] using the Edmonds-Gallai structure theorem.)

3.3 More Examples

Clearly, common hyperedges of an arbitrary number of hereditary hypergraphs on the same set of vertices form a hereditary hypergraph. If all of them arise as stable sets of graphs, the intersection will be just the stable-set-hypergraph of the graph which is the union of the considered graphs. However, if the considered hypergraphs arise in different ways, the intersections may provide nontrivial new cases, if the role of the edges is kept in mind.

More generally, a stable-set in a (not necessarily hereditary) hypergraph is a set so that does not contain any . (Independent sets of matroids are those that do not contain a hyperedge from the circuit-hypergraph.) The family of all stable sets is obviously a hereditary family; , if and only if is a transversal or blocker of the hyperedges; the family of transversals is an upper hereditary hypergraph, another source of examples:

In upper hereditary hypergraphs the supersets of hyperedges are also hyperedges. The dual of is , where . The dual of a hereditary hypergraph is upper hereditary and vice versa, generating more examples; . Each example of upper hereditary hypergraphs provides an example of hereditary hypergraphs, and vice versa. Upper hereditary hypergraphs arise for instance from vertex-sets of graphs that do contain one of a fixed set of graphs as induced subgraphs; being non-planar or non-bipartite is a special case.

In multi (di)graphs with for instance edge-multipicities and we may consider the hereditary hypergraph , when Corollary 3.2 is again meaningful. The upper bound can be replaced by any monoton function of and the graph, combined with vertex multiplicities or edge- and vertex-colored graphs,

3.4 Algorithms and Complexity

The focus of the examples of the previous subsections was the Theorem. Algorithmic and complexity questions are less “choosy” and become meaningful and nontrivial for more examples.

Once in a while questions about particular, critical hereditary hypergraphs are raised anew, sometimes as open problems like in [2] about partitioning the vertex-set into acyclic digraphs. How can the NP-hard covering participate in well-characterizing minmax theorems? The discussion of this question is beyond the possibilities of this note. This will be layed out in forthcoming papers, we mention the key to the solution only shortly:

It is NP-hard to compute and in hereditary hypergraphs it is not easier, since taking the hereditary closure does not affect ! The covering problem for the hereditary closure of -uniform hypergraphs contains the -dimensional matching problem [7], and is therefore NP-hard even if the hyperedges of the hereditary hypergraph are given explicitly, and their number is polynomial in the input size. Indeed, if and only if there exists a partition into triangles.

However, the maximum of the number of vertices covered by non-singletons in a cover of a hereditary hypergraph can be maximized in polynomial time, and the vertex-weighted generalization can also be solved! It can be seen that this maximum does not change if we write here “minimum cover” instead of “cover”. This allows to handle with well characterizing minmax theorems and in polynomial time some aspects of minimum covers [8], for which results of Bouchet [1], Cornuéjols, Hartvigsen and Pulleyblank [3], [4] play an enlightening role.

3.5 Conclusion:

We tried to show by the Theorem and multiple examples how results on graph colorings may be extended to covers in hypergraphs. We continue this work with minmax and structure theorems, develop algorithms at the general level of hereditary hypergraphs, and show more applications and connections between various problems [8], [9]. We hope the reader will also have the reflex of using hereditary hypergraphs when a new special case is coming up!

References

• [1] A. Bouchet, Greedy Algorithm and Symmetric Matroids, Math. Prog. 38, 147–159 (1987).
• [2] J. Bang-Jensen, T. Bellitto, T. Schweser, and M. Stiebitz, Hajós and Ore constructions for digraphs, arXiv:1908.0.
• [3] G. Cornuéjols, W. R. Pulleyblank, Critical Graphs, Matchings and Tours or a Hierarchy of Relaxations for the Travelling Salesman Problem, Combinatorica, 3(1) (1983), 36–52.
• [4] G. Cornuéjols, D. Hartvigsen, W.  R .Pulleyblank, Packing Subgraphs in a Graph, Operations Research letters, Volume 1, Number 4 (1982)
• [5] T. Gallai. Neuer Beweis eines Tutte’schen Satzes. A Magyar Tudományos Akadémia — Matematikai Kutató Intézetének Közleményei, 8:135–139, 1963.
• [6] T. Gallai. Kritische Graphen II, A Magyar Tudományos Akadémia — Matematikai Kutató Intézetének Közleményei, 8:373–395, 1963.
• [7] Garey, Johnson, Computers and Intractability, W. H. Freeman, San Francisco (1979)
• [8] A. Sebő, Minmax theorems and Algorithms for Hereditary Hypergraphs, in preparation.
• [9] A. Sebő, M.  Stehlík, “Matching and Covering in Hereditary Hypergraphs”, manuscript 2009-2017, in preparation.
• [10] M.  Stehlík, Critical graphs with connected complements, J. of Comb. Theory, Series B 89 (2003) 189–194.
• [11] M.  Stehlík, Private communication (October 2019).