\ell-covering k-hypergraphs are quasi-eulerian
aa r X i v : . [ m a t h . C O ] J a n ℓ -covering k -hypergraphs are quasi-eulerian Mateja ˇSajna ∗ and Andrew Wagner University of Ottawa
January 28, 2021
Abstract An Euler tour in a hypergraph H is a closed walk that traverses eachedge of H exactly once, and an Euler family is a family of closed walks thatjointly traverse each edge of H exactly once. An ℓ -covering k -hypergraph , for2 ≤ ℓ < k , is a k -uniform hypergraph in which every ℓ -subset of vertices lietogether in at least one edge.In this paper we prove that every ℓ -covering k -hypergraph, for k ≥ Keywords: ℓ -covering hypergraph; Euler family; Euler tour; Lovasz’s ( g, f )-factor theorem. The complete characterization of graphs that admit an Euler tour is a classic resultcovered by any introductory graph theory course. The concept naturally extends tohypergraphs; that is, an Euler tour of a hypergraph is a closed walk that traversesevery edge exactly once. However, the study of eulerian hypegraphs is a much newerand largely unexplored territory.The first results on Euler tours in hypergraphs were obtained by Lonc and Naroski[4]. Most notably, they showed that the problem of existence of an Euler tour is ∗ Corresponding author. Email: [email protected]. Mailing address: Department of Math-ematics and Statistics, University of Ottawa, 150 Louis-Pasteur Private, Ottawa, ON, K1N 6N5,Canada. k -uniform hypergraphs, for any k ≥
3, as well as whenrestricted to a particular subclass of 3-uniform hypergraphs.Bahmanian and ˇSajna [2] attempted a systematic study of eulerian properties ofgeneral hypergraphs; some of their techniques and results will be used in this paper.In particular, they introduced the notion of an
Euler family — a collection of closedwalks that jointly traverse each edge exactly once — and showed that the problemof existence of an Euler family is polynomial on the class of all hypergraphs.In this paper, we define an ℓ -covering k -hypergraph , for 2 ≤ ℓ < k , to be a non-empty k -uniform hypergraph in which every ℓ -subset of vertices appear together in at leastone edge.In [2], the authors proved that every 2-covering 3-hypergraph with at least two edgesadmits an Euler family, and the present authors gave a short proof [6] to show thatevery triple system — that is, a 3-uniform hypergraph in which every pair of verticeslie together in the same number of edges — admits an Euler tour as long as it has atleast two edges. Most recently, the present authors proved the following result. Theorem 1.1. [7] Let k ≥ , and let H be a ( k − -covering k -hypergraph. Then H admits an Euler tour if and only if it has at least two edges. In this paper, we aim to extend Theorem 1.1 to all ℓ -covering k -hypergraphs. Ourmain result is as follows. Theorem 1.2.
Let ℓ and k be integers, ≤ ℓ < k , and let H be an ℓ -covering k -hypergraph. Then H admits an Euler family if and only if it has at least two edges. As the concept of an Euler family is a relaxation of the concept of an Euler tour, theconclusion of Theorem 1.2 is weaker than that of Theorem 1.1; however, it holds fora much larger class of hypergraphs.We prove Theorem 1.2 by induction on ℓ . The base case ℓ = 2 is stated as Theo-rem 5.1; its proof is essentially a counting argument and requires most of the work.The main part of the proof is presented in Section 5, while some special cases and2echnical details are handled in Sections 3 and 4. In particular, in Section 4, usingthe Lovasz ( g, f )-factor theorem, we develop a sufficient condition for a k -uniformhypergraph without cut edges to admit an Euler family. We use hypergraph terminology established in [1, 2], which applies to loopless graphsas well. Any graph theory terms not explained here can be found in [3].A hypergraph H is a pair ( V, E ), where V is a non-empty set, and E is a multiset ofelements from 2 V . The elements of V = V ( H ) and E = E ( H ) are called the vertices and edges of H , respectively. The order of H is | V | , and the size is | E | . A hypergraphof order 1 is called trivial , and a hypergraph with no edges is called empty .Distinct vertices u and v in a hypergraph H = ( V, E ) are called adjacent (or neigh-bours ) if they lie in the same edge, while a vertex v and an edge e are said to be incident if v ∈ e . The degree of v in H , denoted deg H ( v ), is the number of edges of H incident with v . An edge e is said to cover the vertex pair { u, v } if { u, v } ⊆ e . Ahypergraph H is called k -uniform if every edge of H has cardinality k . Definition 2.1.
Let ℓ and k be integers, 2 ≤ ℓ < k . An ℓ -covering k -hypergraph is a k -uniform hypergraph in which every ℓ -subset of vertices lie together in at least oneedge.The incidence graph of a hypergraph H = ( V, E ) is a bipartite simple graph G withvertex set V ∪ E and bipartition { V, E } such that vertices v ∈ V and e ∈ E of G are adjacent if and only if v is incident with e in H . The elements of V and E arecalled v-vertices and e-vertices of G , respectively.A hypergraph H ′ = ( V ′ , E ′ ) is called a subhypergraph of the hypergraph H = ( V, E )if V ′ ⊆ V and E ′ = { e ∩ V ′ : e ∈ E ′′ } for some submultiset E ′′ of E . For e ∈ E ,the symbol H \ e denotes the subhypergraph ( V, E − { e } ) of H , and for v ∈ V , thesymbol H − v denotes the subhypergraph ( V − { v } , E ′ ) where E ′ = { e − { v } : e ∈ E, e − { v } 6 = ∅} . 3 ( v , v k ) -walk in H is a sequence W = v e v e . . . e k v k such that v , . . . , v k ∈ V ; e , . . . , e k ∈ E ; and v i − , v i ∈ e i with v i − = v i for all i = 1 , . . . , k . A walk is saidto traverse each of the vertices and edges in the sequence. The vertices v , v , . . . , v k are called the anchors of W . If e , e , . . . , e k are pairwise distinct, then W is calleda trail ( strict trail in [1, 2]); if v = v k and k ≥
2, then W is closed .A hypergraph H is connected if every pair of vertices are connected in H ; that is, iffor any pair u, v ∈ V ( H ), there exists a ( u, v )-walk in H . A connected component of H is a maximal connected subhypergraph of H without empty edges. The numberof connected components of H is denoted by c( H ). We call v ∈ V ( H ) a cut vertex of H , and e ∈ E ( H ) a cut edge of H , if c( H − v ) > c( H ) and c( H \ e ) > c( H ),respectively.An Euler family of a hypergraph H is a collection of pairwise anchor-disjoint andedge-disjoint closed trails that jointly traverse every edge of H , and an Euler tour isa closed trail that traverses every edge of H . A hypergraph that is either empty oradmits an Euler tour (family) is called eulerian (quasi-eulerian) . Note that an Eulertour corresponds to an Euler family of cardinality 1, so every eulerian hypergraph isalso quasi-eulerian.The following theorem allows us to determine whether a hypergraph is eulerian orquasi-eulerian from its incidence graph. Theorem 2.2. [2, Theorem 2.18]
Let H be a hypergraph and G its incidence graph.Then the following hold. (1) H is quasi-eulerian if and only if G has a spanning subgraph G ′ such that deg G ′ ( e ) = 2 for all e ∈ E ( H ) , and deg G ′ ( v ) is even for all v ∈ V ( H ) . (2) H is eulerian if and only if G has a spanning subgraph G ′ with at most onenon-trivial connected component such that deg G ′ ( e ) = 2 for all e ∈ E ( H ) , and deg G ′ ( v ) is even for all v ∈ V ( H ) . Technical Lemmas
In this section, we take care of some special cases and prove some technical resultsthat will aid in the proof of our base case, Theorem 5.1.
Lemma 3.1.
Let k ≥ , and let H be a 2-covering k -hypergraph with at least 2 edges.Then H has no cut edges.Proof. Suppose e is a cut edge of H . Then there exist vertices u, v ∈ e that aredisconnected in H \ e . Since H has at least 2 edges, it must be that k = | V ( H ) | and e = V ( H ). Hence there exists w ∈ V ( H ) − e . Let e , e be edges of H containing u and w , and v and w , respectively. As e
6∈ { e , e } , we can see that ue we v is a( u, v )-walk in H \ e , a contradiction. Lemma 3.2.
Let k ≥ , and let H be a 2-covering k -hypergraph of order n > k andsize m ≥ . Then m ≥ ⌊ n +3 k ⌋ .Proof. If n ≤ k −
4, then 2 ⌊ n +3 k ⌋ ≤ ≤ m . Hence assume n ≥ k − n ≥ k −
3. Since there are (cid:0) n (cid:1) pairs of vertices to cover, and eachedge covers (cid:0) k (cid:1) pairs, we know that m ≥ n ( n − k ( k − . As k ≥
4, we have m ≥ n ( n − k ( k − ≥ (3 k − n − k ( k −
1) = 3( n − k = 2 n + n − k ≥ n + 3 k − k ≥ n + 6 k ≥ j n + 3 k k . Finally, assume 2 k − ≤ n ≤ k −
4. As 2 ⌊ n +3 k ⌋ ≤
4, it suffices to show that m ≥ m ≤
3. Since H is a 2-covering k -hypergraph with n > k and m ≥
2, everyvertex has degree at least 2. Thus2 n ≤ X v ∈ V ( H ) deg( v ) = km ≤ k and n ≤ k , contradicting the assumption that n > k .Therefore, in all cases we have m ≥ ⌊ n +3 k ⌋ . emma 3.3. Let H be a hypergraph with | E ( H ) | ≥ satifying the following. • For all e, f ∈ E ( H ) , we have | e ∩ f | ≥ ; and • there exist distinct e, f ∈ E ( H ) such that | e ∩ f | ≥ .Then H is eulerian.Proof. Let E ( H ) = { e , . . . , e m } and assume e and e m are distinct edges such that | e ∩ e m | ≥
3. Take any v ∈ e ∩ e . For i = 2 , . . . , m −
1, let v i be a vertex in( e i ∩ e i +1 ) − { v i − } , and let v ∈ ( e ∩ e m ) − { v , v m − } . It is easy to verify that v e v . . . v m − e m v is an Euler tour of H . Corollary 3.4.
Let H be a 2-covering k -hypergraph of order n . If n ≤ k − or ( k, n ) = (4 , , then H is eulerian.Proof. If n ≤ k −
3, then every pair of edges e, f ∈ E ( H ) satisfies | e ∩ f | ≥
3, so H is eulerian by Lemma 3.3.Assume now that ( k, n ) = (4 , e, f ∈ E ( H ), we have | e ∩ f | ≥
2. Ifthere exist distinct edges e, f ∈ E ( H ) such that | e ∩ f | ≥
3, then H is eulerianby Lemma 3.3. Hence assume | e ∩ f | = 2 for all e, f ∈ E ( H ), and let V ( H ) = { v , . . . , v } . It is not difficult to see that we must have E ( H ) = { e , e , e } where,without loss of generality, the edges are e = v v v v , e = v v v v , and e = v v v v . It follows that W = v e v e v e v is an Euler tour of H . Lemma 3.5.
Let n, k, q ∈ Z + be such that n ≥ qk . Let S = (cid:8) ( x , . . . , x q ) ∈ ( Z + ) q : x + · · · + x q = n, x i ≥ k for all i (cid:9) , and define f : S → Z + by f ( x , . . . , x q ) = (cid:0) x (cid:1) + · · · + (cid:0) x q (cid:1) . Then f attains itsmaximum on S at the point (cid:0) k, . . . , k, n − k ( q − (cid:1) .Proof. Since the domain S is finite, function f indeed attains a maximum on S .6et x = ( x , . . . , x q ) ∈ S be such that f ( x ) is maximum. By symmetry of f , we mayassume that x ≤ x ≤ . . . ≤ x q . As x ≥ k and x q = n − ( x + . . . + x q − ), weobserve that x q ≤ n − k ( q − x q < n − k ( q − i ∈ { , . . . , q − } such that x i > k .Let i be the smallest index with this property, and let y = ( x , . . . , x i − , x i − , x i +1 , . . . , x q − , x q + 1) . Then y ∈ S and f ( y ) = q − X j =1 j = i (cid:18) x j (cid:19) + (cid:18) x i − (cid:19) + (cid:18) x q + 12 (cid:19) = q − X j =1 j = i (cid:18) x j (cid:19) + x i ( x i − − x i − x q ( x q − x q q X j =1 (cid:18) x j (cid:19) + ( x q − x i + 1) > f ( x ) , contradicting the choice of x .Hence x q = n − k ( q − x = . . . = x q − = k . Thus f attains itsmaximum on S at the point x = (cid:0) k, . . . , k, n − k ( q − (cid:1) as claimed. In this section, we state and prove Proposition 4.2, which gives a sufficient conditionfor a k -uniform hypergraph to admit an Euler family. This sufficient condition willbe our main tool in the proof of Theorem 5.1. It is based on the ( g, f )-factor theoremby Lov´asz [5], stated below as Theorem 4.1.For a graph G and functions f, g : V ( G ) → N , a ( g, f ) -factor of G is a spanningsubgraph F of G such that g ( x ) ≤ deg F ( x ) ≤ f ( x ) for all x ∈ V ( G ). An f -factor issimply an ( f, f )-factor. For any sets U, W ⊆ V ( G ), let ε G ( U, W ) denote the numberof edges of G with one endpoint in U and the other in W .7 heorem 4.1. [5] Let G = ( V, E ) be a graph and f, g : V → N be functions suchthat g ( x ) ≤ f ( x ) and g ( x ) ≡ f ( x ) (mod 2) for all x ∈ V . Then G has a ( g, f ) -factor F such that deg F ( x ) ≡ f ( x ) (mod 2) for all x ∈ V if and only if, for all disjoint S, T ⊆ V , we have X x ∈ S f ( x ) + X x ∈ T (deg G ( x ) − g ( x )) − ε G ( S, T ) − q ( S, T ) ≥ , (1) where q ( S, T ) is the number of connected components C of G − ( S ∪ T ) such that X x ∈ V ( C ) f ( x ) + ε G ( V ( C ) , T ) ≡ (mod 2). Proposition 4.2.
Let k ≥ , and let H = ( V, E ) be a k -uniform hypergraph of order n and size m . Let G be the incidence graph of H , and G ∗ the graph obtained from G by appending m + n ) loops to every v-vertex.Assume that H has no cut edges and that for all X ⊆ E with | X | ≥ , we have that | X | ≥ ⌊ c( G ∗ − X )+3 k ⌋ . Then H is quasi-eulerian.Proof. Let r = 2( m + n ) , and define f : V ( G ∗ ) → Z by f ( x ) = (cid:26) r if x ∈ V, x ∈ E. We shall use Theorem 4.1 to show that G ∗ has an ( f, f )-factor, so let S, T ⊆ V ( G ∗ )be disjoint sets, and denote γ ( S, T ) = X x ∈ S f ( x ) + X x ∈ T (deg G ∗ ( x ) − f ( x )) − ε G ∗ ( S, T ) − q ( S, T ) , where q ( S, T ) is the number of connected components C of G ∗ − ( S ∪ T ) such that ε G ∗ ( V ( C ) , T ) is odd. Observe that Condition (1) for G ∗ with g = f is equivalent to γ ( S, T ) ≥ G is a subgraph of K n,m , we have ε G ∗ ( S, T ) ≤ mn and q ( S, T ) ≤ m + n , andtherefore ε G ∗ ( S, T )+ q ( S, T ) ≤ ( m + n ) = r . In addition, we have deg G ∗ ( x ) − f ( x ) ≥ r for all x ∈ V , and deg G ∗ ( x ) − f ( x ) ≥ k − x ∈ E .8 ase 1: ( S ∪ T ) ∩ V = ∅ . If S ∩ V = ∅ , then X x ∈ S f ( x ) ≥ r , and if T ∩ V = ∅ , then X x ∈ T (deg G ∗ ( x ) − f ( x )) ≥ r . Thus, in both cases γ ( S, T ) = (cid:16) X x ∈ S f ( x ) + X x ∈ T (deg G ∗ ( x ) − f ( x )) (cid:17) − (cid:16) ε G ∗ ( S, T ) + q ( S, T ) (cid:17) ≥ r − r ≥ . Case 2: ( S ∪ T ) ∩ V = ∅ . Then ε G ∗ ( S, T ) = 0 since S ∪ T ⊆ E .First, suppose T = ∅ . Then ε G ∗ ( V ( C ) , T ) = 0 for all connected components C of G ∗ − ( S ∪ T ), so q ( S, T ) = 0. Hence γ ( S, T ) = X x ∈ S f ( x ) ≥ S = ∅ and | T | = 1. Then S ∪ T = { e } for some e ∈ E . By assumption,edge e is not a cut edge of H and hence by [1, Theorem 3.23], e-vertex e is not a cutvertex of G ∗ , and G ∗ − ( S ∪ T ) is connected. It follows that q ( S, T ) ≤ γ ( S, T ) = (deg G ∗ ( e ) − f ( e )) − q ( S, T ) ≥ ( k − − ≥ . We may now assume that T = ∅ and | S ∪ T | ≥
2. Since each connected component C of G ∗ − ( S ∪ T ) with ε G ∗ ( V ( C ) , T ) odd corresponds to at least one edge incidentwith a vertex in T , the number of such components is at most k | T | . Hence q ( S, T ) ≤ min { c( G ∗ − ( S ∪ T )) , k | T |} , and γ ( S, T ) = 2 | S | + ( k − | T | − q ( S, T ) ≥ | S ∪ T | + ( k − | T | − min { c( G ∗ − ( S ∪ T )) , k | T |} . (2)Define t = ⌊ c( G ∗ − ( S ∪ T ))+3 k ⌋ , so that kt − ≤ c( G ∗ − ( S ∪ T )) ≤ kt + k − . If | T | ≥ t + 1, then min { c( G ∗ − ( S ∪ T )) , k | T |} = c( G ∗ − ( S ∪ T )) ≤ kt + k −
4, soInequality (2) yields γ ( S, T ) ≥ | S ∪ T | + ( k − t + 1) − ( kt + k −
4) = 2 | S ∪ T | − t. | T | ≤ t : in this case, we have min { c( G ∗ − ( S ∪ T )) , k | T |} ≤ k | T | , so that (2) yields γ ( S, T ) ≥ | S ∪ T | + ( k − | T | − k | T | = 2 | S ∪ T | − | T | ≥ | S ∪ T | − t. In both cases, as S ∪ T ⊆ E and | S ∪ T | ≥
2, the assumption of the propositionimplies | S ∪ T | ≥ ⌊ c( G ∗ − ( S ∪ T ))+3 k ⌋ = 2 t , so that γ ( S, T ) ≥ γ ( S, T ) ≥ S, T ⊆ V ( G ∗ ), and by Theorem 4.1, weconclude that G ∗ has an ( f, f )-factor F . Deleting the loops of F , we obtain aspanning subgraph F ′ of G in which all v-vertices have even degree and all e-verticeshave degree 2. Thus H admits an Euler family by Theorem 2.2. We shall now prove our main result, Theorem 1.2. We use induction on ℓ , and mostof the work is required to prove the basis of induction, which we state below asTheorem 5.1. Theorem 5.1.
Let k ≥ , and let H be a 2-covering k -hypergraph with at least twoedges. Then H is quasi-eulerian.Proof. Let H = ( V, E ) with n = | V | and m = | E | . If n ≤ k −
3, then H is eulerianby Corollary 3.4, so we may assume that n ≥ k − . If n ≤ k , it then follows that ( k, n ) = (4 , H is eulerian by Corollary 3.4.Hence n > k , and Lemma 3.2 implies that m ≥ (cid:4) n +3 k (cid:5) .In the rest of the proof we show that H satisfies the conditions of Proposition 4.2.Let G ∗ be the graph obtained from the incidence graph of H by adjoining 2( m + n ) loops to every v-vertex.Fix any X ⊆ E with | X | ≥
2, and denote q = c( G ∗ − X ).10uppose that | X | < j q +3 k k . If q ≤ k −
4, then this supposition implies that | X | < q ≥ k −
3, and hence q ≥
5. Moreover,our supposition implies | X | ≤ q + 3 k − . (3)Let ℓ denote the number of v-vertices that are isolated in G ∗ − X . Case 1: ℓ ≥ . If ℓ = n , then X = E , q = n , and | X | = | E | ≥ ⌊ n +3 k ⌋ = 2 ⌊ q +32 ⌋ ,contradicting our assumption on X . Thus we may assume ℓ < n , and hence ℓ < q .Since G ∗ − X has q − ℓ non-trivial connected components, each with at least k v-vertices, we have n ≥ ℓ + k ( q − ℓ ) . (4)Since q > ℓ , this inequality also implies n ≥ ℓ + k. (5)Let S be the set of pairs { u, v } of v-vertices such that u is isolated in G ∗ − X , and v is not. Then | S | = ℓ ( n − ℓ ). Observe that every edge of H covers at most k pairsfrom S , which implies that | X | ≥ ℓ ( n − ℓ ) k . Combining this inequality with (3), weobtain 4 ℓ ( n − ℓ ) k ≤ q + 6 − kk . (6)Substituting q ≤ ℓ + n − ℓk from Inequality (4) and rearranging yields n (4 ℓ − ≤ ℓ − k + 2 ℓk − ℓ + 6 k. Further substituting n ≥ ℓ + k from (5) and isolating ℓ , we obtain ℓ ≤ − k , whichimplies ℓ ∈ { , } as k ≥ n − ℓk ≥ q − ℓ from (4) andsimplify, then we obtain (4 ℓ − q − ℓ ≤ − k ≤ . ℓ = 1 or ℓ = 2 yields q ≤
3, a contradiction.
Case 2: ℓ = 0 . Let C , C , . . . , C q be the connected components of G ∗ − X , and let n i denote the number of v-vertices of C i . Note that n i ≥ k for all i .The number of pairs of v-vertices that lie in distinct connected components of G ∗ − X is (cid:0) n (cid:1) − P qi =1 (cid:0) n i (cid:1) , and these pairs must all be covered by the edges of X . As n ≥ qk , n + . . . + n q = n , and n i ≥ k , for all i ,we know that P qi =1 (cid:0) n i (cid:1) ≤ ( q − (cid:0) k (cid:1) + (cid:0) n − k ( q − (cid:1) by Lemma 3.5. Therefore, (cid:18) n (cid:19) − q X i =1 (cid:18) n i (cid:19) ≥ (cid:18) n (cid:19) − ( q − (cid:18) k (cid:19) − (cid:18) n − k ( q − (cid:19) . Since each edge of X covers up to (cid:0) k (cid:1) pairs of v-vertices in distinct connected com-ponents, we deduce that | X | ≥ (cid:0) n (cid:1) − ( q − (cid:0) k (cid:1) − (cid:0) n − k ( q − (cid:1)(cid:0) k (cid:1) . On the other hand, by (3), we have | X | ≤ q +6 − kk , so (cid:0) n (cid:1) − ( q − (cid:0) k (cid:1) − (cid:0) n − k ( q − (cid:1)(cid:0) k (cid:1) ≤ q + 6 − kk . (7)We now substitute x = q −
1, noting that x ≥ q ≥
5. Rearranging Inequality(7), we then obtain2 kxn ≤ k x + ( k + 2 k − x − ( k − k − . Applying n ≥ qk = ( x + 1) k further yields k x + ( k − k + 2) x + ( k − k − ≤ . Denote the left-hand side by f ( x ) = ax + bx + c , where a = k , b = k − k + 2,and c = ( k − k − a, b > k ≥
4. If b − ac <
0, then f ( x ) > x , a contradiction. Hence assume b − ac ≥
0. Let x be the larger12f the two roots of f ( x ) = 0. If x <
4, then f ( x ) > x ≥
4, a contradiction.Hence we must have 4 ≤ − b + √ b − ac a . Since a, b >
0, it is straightforward to show that 16 a + 4 b + c ≤ a + 4 b + c = k (21 k −
17) + 16 > , a contradiction.Since each case leads to a contradiction, we conclude that | X | ≥ ⌊ c( G ∗ − X )+3 k ⌋ . ByLemma 3.1, hypergraph H has no cut edges, so we may apply Proposition 4.2 toconclude that H is quasi-eulerian.We are now ready to prove our main result, restated below. Theorem 1.2.
Let ℓ and k be integers, ≤ ℓ < k , and let H be an ℓ -covering k -hypergraph. Then H is quasi-eulerian if and only if it has at least two edges.Proof. Since H is non-empty, and since a hypergraph with a single edge does notadmit a closed trail, necessity is easy to see.To prove sufficiency, for s ≥ ℓ ≥
2, define the proposition P s ( ℓ ) : “Every ℓ -covering ( ℓ + s )-hypergraph with at least two edges is quasi-eulerian.”Theorem 1.1 implies that P ( ℓ ) holds for all ℓ ≥
2. Hence fix any s ≥ P s ( ℓ ) by induction on ℓ . As ℓ + s ≥
4, the basis of induction, P s (2), followsfrom Theorem 5.1. Suppose that, for some ℓ ≥
2, the proposition P s ( ℓ ) holds; thatis, every ℓ -covering ( ℓ + s )-hypergraph with at least two edges is quasi-eulerian.Let H = ( V, E ) be an ( ℓ + 1)-covering (cid:0) ( ℓ + 1 (cid:1) + s )-hypergraph with | E | ≥ . Fixany v ∈ V and let V ∗ = V − { v } . Define a mapping ϕ : E → V ∗ by ϕ ( e ) = e − { v } if v ∈ e, ϕ ( e ) = e − { u } for any u ∈ e. Then let E ∗ = { ϕ ( e ) : e ∈ E } and H ∗ = ( V ∗ , E ∗ ), so that ϕ is a bijection from E to E ∗ . It is straightforward to verify that H ∗ is an ℓ -covering ( ℓ + s )-hypergraph.As | E ∗ | = | E | ≥
2, by induction hypothesis, hypergraph H ∗ admits an Euler family F ∗ . In each closed trail in F ∗ , replace each e ∈ E ∗ with ϕ − ( e ) to obtain a set F of closed trails of H . It is not difficult to verify that F is an Euler family of H , so P s ( ℓ + 1) follows.By induction, we conclude that P s ( ℓ ) holds for all ℓ ≥
2, and any s ≥
1. Therefore,every ℓ -covering k -hypergraph with at least two edges is quasi-eulerian. Acknowledgements
The first author gratefully acknowledges support by the Natural Sciences and EngineeringResearch Council of Canada (NSERC), Discovery Grant RGPIN-2016-04798.
References [1] M. A. Bahmanian and M. ˇSajna, Connection and separation in hypergraphs,
TheoryAppl. Graphs (2015), Art. 5, 24 pp.[2] M. A. Bahmanian and M. ˇSajna, Quasi-Eulerian hypergraphs, Elec. J. Combin. (2017), Graph Theory , Springer, 2008.[4] Z. Lonc and P. Naroski, On tours that contain all edges of a hypergraph,
Elec. J.Combin. (2010), Acta Math. Acad. Sci. Hungar. (1972),223–246.[6] M. ˇSajna and A. Wagner, Triple systems are eulerian, J. Combin. Des. (2017),185–191.[7] M. ˇSajna and A. Wagner, Covering hypergraphs are eulerian , submitted, 24 pp.arXiv:2101.04561., submitted, 24 pp.arXiv:2101.04561.