# Ear-Slicing for Matchings in Hypergraphs

We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a 3-regular, 3-uniform hypergraph into at most one triangle (hyperedge of size 3) and edges (subsets of size 2 of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of Cornuéjols' and Pulleyblank's. The existence of such a packing can be considered to be a hypergraph variant of Petersen's theorem on perfect matchings, and leads to a simple proof for a sharpening of Lu's theorem on antifactors of graphs.

## Authors

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## 1 Introduction

Given a hypergraph , where is the power-set of we will call the elements of vertices, and those of hyperedges, . We say that a hypergraph is -uniform if all of its hyperedges have elements, and it is -regular if all of its vertices are contained in hyperedges. Hyperedges may be present with multiplicicities, for instance the hypergraph with consisting of the hyperedge with multiplicity is a -uniform, -regular hypergraph. The hereditary closure of the hypergraph is where , and is hereditary, if . For the new edges of the hereditary closure we do not need to define multiplicities we will consider them all to be one.

Hyperedges of cardinality will be called singletons, those of cardinality and are called edges and triangles respectively. Deleting a vertex of the hypergraph results in the hypergraph . For hereditary hypergraphs this is the same as deleting from all hyperedges. The degree of in is the number of hyperedges containing .

Given a hypergraph , denote by the set of edges (of size two) in , . We do not need parallel edges in , we suppose is a graph without parallel edges or loops. The (connected) components of are defined as 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 is also called component. We define a graph as a hypergraph with , that is, a -uniform hypergraph without loops or parallel edges.

A matching in a graph is a set of pairwise vertex-disjoint edges. A matching is perfect if it partitions the vertex-set of the graph. A graph is called factor-critical if has a perfect matching (also called a -factor) for all .

In this note we prove two lemmas, possibly interesting for their own sake, on when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph, leading to a result on -uniform hypergraphs (Section 2). We then prove that a -regular and -uniform hypergraph is perfectly matchable in some sense (a generalization of Petersen’s theorem [6] on -uniform hypergraphs), sharpening a result of Lu’s [5] (Section 3).

## 2 Ears and Triangles

An ear-decomposition consists of a circuit , and paths sharing its (one or two) endpoints with ; are called ears. An ear is called trivial, if it consists of one edge. An ear is called odd

if it has an odd number of edges. Lovász

[3], [4] proved that a graph is factor-critical if and only if it has an ear-decomposition with all ears odd.

For , denote by ear the index of the first ear when vertex occurs. (It may occur later only as an endpoint of an ear.) Given an ear-decomposition, we call an edge odd, if earear, and if we also require that is joined to an endpoint of by an odd subpath of not containing , and that the same holds interchanging the role of and . We will call an odd ear-decomposition maximal if for every odd edge , earear, we have .

Clearly, there exists a maximal odd ear-decomposition, since while there are odd edges with endpoints on , we can obviously replace the ears and , where is necessarily a trivial ear, with two odd ears. In particular, an odd ear-decomposition with a maximum number of nontrivial ears (equivalently, with a minimum number of trivial ears) is maximal.

We need the following lemmas that may also have some self-interest and other applications: for , , it provides a sufficient condition for to have a perfect matching containing .

###### Lemma 2.1.

Let be a factor-critical graph given with an odd ear-decomposition, and let be an odd edge in a nontrivial ear. For any vertex with earearear, there exists a perfect matching of containing .

Proof : Let be the ear-decomposition. We can suppose without loss of generality (since by the easy direction of Lovász’s theorem [3], a graph having an odd ear-decomposition is factor-critical) that is on the last ear . Since is in , and is factor-critical (again by the easy direction of Lovász’s theorem), has a perfect matching . Adding the odd edges of to , we get the matching of the assertion.

Cornuéjols, Hartvigsen and Pulleyblank [1], [2] (see also [4]) need to check when a factor-critical graph is partitionable into triangles and edges, and for this they try out all triangles. The following lemma improves this for the unions of triangles by showing that they always have such a partition:

###### Lemma 2.2.

If is a -uniform hypergraph and is factor-critical, then has a partition into one triangle and a perfect matching in .

Proof : Consider a maximal odd ear-decomposition, let be an odd edge on its last nontrivial ear and let be a triangle containing . If ear, we are done by Lemma 2.1: a matching , of and the triangle do partition . Suppose now ear.

If choose a vertex on at odd distance from both and , and let both and denote this same vertex. The following proof holds then for both or .

We can suppose without loss of generality that the endpoint of the ear , , , and the other endpoint of (possibly ) follow one another in this order on the ear. The path between and is even, because if it were odd, the edge would be odd - the path between and being odd by the assumption that the edge is odd -, contradicting the maximality of the ear-decomposition. But then a perfect matching of and every second edge of the subpath of between and , covering but not covering , and the odd edges of the rest of including , form a perfect matching in containing . Replacing in this perfect matching by finishes the proof.

## 3 Regular Hypergraphs

###### Theorem 3.1.

If is a -uniform, -regular hypergraph, then has either a perfect matching (if is even), or it is factor-critical (if is odd) and in the latter case can be partitioned into one triangle and a perfect matching of .

Proof : If has a perfect matching we have nothing to prove. Suppose it has not.

Claim. is factor-critical.

We prove more: for , , the number of components of satisfies . (Then by Tutte’s theorem [7], see also [4], has a perfect matching for all .) For each component , by -regularity, .

In this sum, divisible by , every hyperedge is counted as many times as it has vertices in . Since is connected, and , we have that the sum of for all hyperedges that meet , itself divisible by by -uniformity, is strictly larger than . Therefore, the sum of for these edges is nonzero and also divisible by , so it is at least . The vertices not in of the edges that meet are in , hence , so summing for all edges, the sum is at least :

 3k≤∑e∈E|e∩X|≤∑x∈XdH(x)=3|X|,

so , finishing the proof of the claim. Now Lemma 2.2 can be readily applied.

The intuition that at most one triangle may be enough is highly influenced by Cornuéjols, Hartvigsen and Pulleyblank’s work [1], [2], even if these are not explicitly used. The heart of the proof is encoded in the two lemmas that show: we can either increase the number of nontrivial ears or find the wanted partition, and for this, -uniformity is not needed. The proof is clearly algorithmic, providing a low degree polynomial algorithm.

Finally, we prove a sharpening of Lu’s theorem [5], which considered a question in [4]. A simple proof of Lu’s theorem has been the initial target of this work.

###### Corollary 3.1.

Let be a -regular bipartite graph with bipartition and . Then has a subgraph with all degrees of vertices in equal to , all degrees of vertices in equal to or , except possibly at most one vertex of which is of degree .

Proof : Delete pairwise disjoint perfect matchings one by one (they are well-known to exist in bipartite regular graphs  [4] by Hall’s theorem, actually a -edge-coloring also exists by Kőnig’s edge-coloring theorem). Define then the hypergraph with , and to have one hyperedge for each consisting of the set of neighbors of . Since there are no loops or parallel edges in (see Section 1), the defined hypergraph is -uniform and -regular.

Apply now Theorem 3.1.

As the proof shows, the essential case is , when the theorem can be considered to be a generalization of Petersen’s theorem [6] about perfect matchings in graphs. Let us also state the reformulation to hypergraphs by the inverse of the correspondence in the proof:

###### Corollary 3.2.

If is a -uniform, -regular hypergraph, , then can be partitioned into hyperedges of of size and at most one hyperedge of size .

Acknowledgment: Many thanks to Zoltán Szigeti and Louis Esperet for precious suggestions!

## References

• [1] 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.
• [2] G. Cornuéjols, D. Hartvigsen, W.  R .Pulleyblank, Packing Subgraphs in a Graph, Operations Research letters, Volume 1, Number 4 (1982)
• [3] L. Lovász, A note on factor-critical graphs, Studia Sci. Math. Hungar., 7, 1972, 279–280.
• [4] L. Lovász, M. D. Plummer, Matching theory, Ann. Discrete Math., 29 North Holland, Amsterdam, 1986.
• [5] H. Lu, Antifactor of regular bipartite graphs, https://arxiv.org/pdf/1511.09277v1.pdf
• [6] J. Petersen, Die Theorie der regulären Graphen, Acta Math,, 15, 1891, 193–220, Jbuch. 23-115.
• [7] W. T. Tutte, The factorization of linear graphs, J. of the London Mathematical Society, 22, 107-111.