Minimal non-odd-transversal hypergraphs and minimal non-odd-bipartite hypergraphs

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are more close to bipartite simple graphs from the viewpoint of both structure and spectrum. A hypergraph is called minimal non-odd-transversal if it is non-odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.


Introduction
Let G = (V, E) be a hypergraph, where V =: V (G) is the vertex set, and E =: E(G) is the edge set whose elements e ⊆ V . If for each edge e of G, |e| = k, then G is called a k-uniform hypergraph. The degree d(v) of a vertex v of G is defined to be the number of edges of G containing v. If d(v) = d for all vertices v of G, then G is called d-regular.
A hypergraph G is called 2-colorable if there exists a 2-coloring of the vertices of V (G) such that G contains no monochromatic edges; and it is called minimal non-2-colorable if it is non-2-colorable but deleting any edge from E(G) results in a 2-colorable hypergraph. Seymour [25] proved that if G is minimal non-2-colorable and V (G) = ∪{e ∈ E(G)}, then |E(G)| ≥ |V (G)|. Aharoni and Linial [1] presented an infinite version of Seymour's result. Alon and Bregman [3] proved that if k ≥ 8 then every k-regular k-uniform hypergraph is 2-colorable. Henninga and Yeoa [14] showed that Alon-Bergman result is true for k ≥ 4.
A subset U of V (G) is called a transversal (also called vertex cover or hitting set ) of G if each edge of G has a nonempty intersection with U . The transversal number of G is the minimum size of transversals in G, which was well studied by Alon [2], Chvátal and McDiarmid [6], Henninga and Yeo [15]. G is called bipartite if for some nonempty proper subset U ⊆ V (G), U and its complement U c are both transversal; or equivalently the vertex set V (G) has a bipartition into two parts such that every edge of E(G) intersects both parts. Surely, G is bipartite if and only if G is 2-colorable.
A subset U of V (G) is called an odd transversal of G if each edge of G intersects U in an odd number of vertices [8,24]. A hypergraph G is called odd-transversal if it has an odd transversal. Nikiforov [20] firstly uses odd transversal to investigate the spectral symmetry of tensors and hypergraphs. Hu and Qi [16] introduce the notion of odd-bipartite hypergraph to study the zero eigenvalue of the signless Laplacian tensor.
Definition 1.1 ( [16]). Let G be a k-uniform hypergraph G, where k is even. If there exists a bipartition {U, U c } of V (G) has such that each edge of G intersects U (and also U c ) in an odd number of vertices, then G is called odd-bipartite, and {U, U c } is an odd-bipartition of G.
So, odd-bipartite hypergraphs are surely odd-transversal hypergraphs and bipartite hypergraphs. For the uniform hypergraphs with even uniformity, the notion of odd-bipartite hypergraphs is equivalent to that of odd-transversal hypergraphs.
From the viewpoint of spectrum, a simple graph is bipartite if and only if its adjacency matrix has a symmetric spectrum. However, the adjacency tensor of a bipartite uniform hypergraph does not possess such property. We note that the hypergraphs under consideration are uniform when discussing their spectra. Shao et al. [26] proved that the adjacency tensor of a k-uniform hypergraph G has a symmetric H-spectrum if and only if k is even and G is odd-bipartite. So, the oddbipartite hypergraphs are more close to bipartite simple graphs than the bipartite hypergraphs based on the following two reasons. First they both have a structural property, namely, there exists a bipartition of the vertex set such that every edge intersects the each part of the bipartition in an odd number of vertices. Second they both have a symmetric H-spectrum.
There are some examples of odd-bipartite hypergraphs, e.g. power of simple graphs and cored hypergraphs [17], hm-hypergraphs [16], m-partite m-uniform hypergraphs [7]. Nikiforov [21] gives two classes of non-odd-transversal hypergraphs. Fan et al. [18] construct non-odd-bipartite generalized power hypergraphs from non-bipartite simple graphs. It is known that a connected bipartite simple graph has a unique bipartition up to isomorphism. However, an odd-bipartite hypergraph can have more than one odd-bipartition. Fan et al. [11] given a explicit formula for the number of odd-bipartition of a hypergraph by the rank of its incidence matrix over Z 2 . So, it seems hard to give examples of non-odd-bipartite hypergraphs.
To our knowledge, there is no characterization of non-odd-transversal or nonodd-bipartite hypergraphs. We observe that non-odd-transversal hypergraphs have a hereditary property, that is, if G contains a non-odd-transversal sub-hypergraph, then G is non-odd-transversal. G is called minimal non-odd-transversal, if G is non-odd-transversal but deleting any edge from G results in an odd-transversal hypergraph, or equivalently, any nonempty proper edge-induced sub-hypergraph of G is odd-transversal. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by the degrees and the rank of its incidence matrix over Z 2 . If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterized 2-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of d-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

Basic notions
Unless specified somewhere, all hypergraphs in this paper contain no multiple edges or isolated vertices, where vertex is called isolated if it is not contained in any edge of the hypergraph. Let G = (V, E) be a hypergraph. G is called square if |V | = |E|. A walk of length t in G is a sequence of alternate vertices and edges: v 0 e 1 v 1 e 2 . . . e t v t , where {v i , v i+1 } ⊆ e i for i = 0, 1, . . . , t − 1. G is said to be connected if every two vertices are connected by a walk.
The vertex-induced sub-hypergraph of G by the a subset U ⊆ V (G), denoted by G| U , is a hypergraph with vertex set U and edge set {e ∩ U : e ∈ E(G), e ∩ U = ∅}. For a connected hypergraph G, a vertex v is called a cut vertex of G if G| V (G)\{v} is disconnected. The edge-induced sub-hypergraph of G by a subset F ⊆ E(G), denoted by G| F , is a hypergraph with vertex set ∪ e∈F e and edge set F .
Let G be a hypergraph and let e be an edge of G. Denote by G−e the hypergraph obtained from G by deleting the edge e from E(G). For a connected hypergraph G, an edge e is called a cut edge of G if G − e is disconnected.
A matching M of G is a set of pairwise disjoint edges of G. In particular, if G is bipartite simple graph with a bipartition The incidence bipartite graph Γ G of G is a bipartite simple graph with two parts V (G) and E(G) such that {v, e} ∈ E(Γ G ) if and only if v ∈ e.
The edge-vertex incidence matrix of G, denoted by B G = (b e,v ), is a matrix of size |E(G)| × |V (G)|, whose entries b e,v = 1 if v ∈ e, and b e,v = 0 otherwise.
The dual of G, denoted by G * , is the hypergraph whose vertex set is E(G) and edge set is {{e ∈ E(G) : v ∈ e} : v ∈ V (G)}. If no two vertices of G are contained in precisely the same edges of G, then (G * ) * is isomorphic to G. In this situation, the incidence bipartite graph Γ G ∼ = Γ G * , and the incidence matrix B G = B ⊤ G * , where the latter denotes the transpose of B G * .
Let G be a simple graph, and let k be even integer greater than 2. Denote by G k, k 2 the hypergraph obtained from G whose vertex set is ∪ v∈V (G) v and edge set {u ∪ v : {u, v} ∈ E(G)}, where v denotes an k 2 -set corresponding to v, and all those sets are pairwise disjoint; intuitively G k, k 2 is obtained from G by blowing up each vertex into an k 2 -set and preserving the adjacency relation [18]. It is proved that G k, k 2 is non-odd-bipartite if and only if G is non-bipartite [18]. Next we will introduce some knowledge of eigenvalues of a tensor. For integers k ≥ 2 and n ≥ 2, a tensor (also called hypermatrix ) T = (t i1...i k ) of order k and dimension n refers to a multidimensional array t i1i2...i k such that t i1i2...i k ∈ C for all i j ∈ [n] := {1, 2, . . . , n} and j ∈ [k]. T is called symmetric if its entries are invariant under any permutation of their indices.
Given a vector x ∈ C n , T x k ∈ C, and T x k−1 ∈ C n , which are defined as follows: Let I be the identity tensor of order k and dimension n, that is, i i1i2...i k = 1 if and only if i 1 = i 2 = · · · = i k ∈ [n] and zero otherwise. , . . . , x k−1 n ) ∈ C n . The characteristic polynomial ϕ T (λ) of T is defined as the resultant of the polynomials (λI − T )x k−1 ; see [23,5,13]. It is known that λ is an eigenvalue of T if and only if it is a root of ϕ T (λ). The spectrum of T is the multi-set of the roots of ϕ T (λ).
Suppose that T is real. If x is a real eigenvector of T , surely the corresponding eigenvalue λ is real. In this case, x is called an H-eigenvector and λ is called an H-eigenvalue. The H-spectrum of T is the set of all H-eigenvalues of T , denote by HSpec(T ). The spectral radius of T is defined as the maximum modulus of the eigenvalues of T , denoted by ρ(T ). Denote by λ max (T ), λ min (T ) the largest H-eigenvalue and the least H-eigenvalue of T , respectively.
For a symmetric tensor, we have the following result.
Lemma 2.2. Let T be a real symmetric tensor of order k and dimension n.
Furthermore, x is an optimal solution of the above optimization if and only if it is an eigenvector of T associated with λ max (T ).
(2) [[23], Theorem 5] If k is also even, then and x is an optimal solution of the above optimization if and only if it is an eigenvector of T associated with λ min (T ).
otherwise. The spectral radius, the least H-eigenvalue of G are referring to its adjacency tensor A(G), denoted by ρ(G), λ min (G) respectively. The H-spectrum of A(G) is denoted by HSpec(G).
The spectral hypergraph theory has been an active topic in algebraic graph theory recently; see e.g. [7,9,10,21,22]. By the Perron-Frobenius theorem for nonnegative tensors [4,12,27,28,29], ρ(G) is exactly the largest H-eigenvalue of A(G). If G is connected, there exists a unique positive eigenvector up to scales associated with ρ(G), called the Perron vector of G. Noting that the adjacency tensor A(G) is nonnegative and symmetric, so ρ(G) holds (1) of Lemma 2.2, and λ min (G) holds (2) of Lemma 2.2 if k is even. By Perron-Frobenius theorem, λ min (G) ≥ −ρ(G). By the following lemma, if G is connected and non-odd-bipartite, then λ min (G) > −ρ(G). [21,20,26,11] Let G be a k-uniform connected hypergraph. Then the following results are equivalent.
(1) k is even and G is odd-bipartite.
Finally, we introduce some notations used throughout out the paper. Denote by C n a cycle of length n as a simple graph. Denote by ½ an all-one vector whose size can be implicated by the context, rankA the rank of a matrix A over Z 2 , and F q a field of order q.

Characterization of minimal non-odd-transversal hypergraphs
In this section we will give some equivalent conditions in terms of degrees and rank of the incidence matrix over Z 2 for a hypergraph to be minimal non-oddtransversal.
Lemma 3.1. If G is a minimal non-odd-transversal hypergraph, then G is connected and contains no cut vertices.
Proof. If G contains more than one connected component, then at least one of them is non-odd-transversal, a contradiction to the definition. So G itself is connected. Suppose G contains a cut vertex. Then G is obtained from two connected nontrivial sub-hypergraphs G 1 , G 2 sharing exactly one vertex (the cut vertex). So, at least one of G 1 , G 2 is non-odd-transversal, also a contradiction.
Lemma 3.2. Let G be a connected hypergraph, and B G be the edge-vertex incidence matrix of G. Then G is odd-transversal if and only if the equation has a solution, or equivalently and χ e (v) = 0 otherwise. Then B G consists of those χ e as row vectors for all e ∈ E(G). Lemma 3.3. Let G be a connected hypergraph with m edges. If m is odd, and each vertex has an even degree, or equivalently e∈E(G) χ e = 0 over Z 2 , then G is non-odd-transversal.
Proof. Let e 1 , . . . , e m be edges of G. Write (B G , ½) as the following form: Adding the first row to all other rows over Z 2 , we will have So, rank(B G , ½) = 1 + rankC. As m is odd and e∈E(G) χ e = 0, implying that rankB G = rankC. By Lemma 3.2, G is non-odd-transversal.
Theorem 3.4. Let G be a connected hypergraph with m edges. The following are equivalent.
(2) m is odd, e∈E(G) χ e = 0 over Z 2 , and e∈F χ e = 0 over Z 2 for any nonempty proper subset F of E(G).
(4) m is odd, each vertex of G has an even degree, and any nonempty proper edge-induced sub-hypergraph of G contains vertices of odd degrees.
We assert that a i = 1 for i = 2, . . . , m. Otherwise, there exists a j, 2 ≤ j ≤ m, such that a j = 0. Then χ e1 is also a linear combination of χ ei + χ e1 for i = 2, . . . , m , which is an even number, a contradiction. So, m is odd, and e∈E(G) χ e = 0 by Eq. (3.5). Assume to the contrary there exists a nonempty proper subset F of E(G), e∈F χ e = 0 over Z 2 . If |F | is odd, then by Lemma 3.3, the sub-hypergraph G| F induced by the edges of F is non-oddtransversal, a contradiction to the definition. Otherwise, |F | is even, then |E(G)\F | is odd as m is odd, and the sub-hypergraph G| E(G)\F is non-odd-transversal, also a contradiction.
(3) ⇒ (1). By Lemma 3.3, G is non-odd-transversal. Let e be an arbitrary edge of G. Adding all rows χ f for f = e to the row χ e will yield a zero row as e∈E(G) χ e = 0. So rankB G−e = rankB G = m−1 over Z 2 , implying that B G−e has full rank over Z 2 with respect to rows. Hence, rankB G−e = rank(B G−e , ½) over Z 2 , and G − e is odd-transversal by Lemma 3.2. So G is minimal non-odd-transversal.
Of course (2) is equivalent to (4).  From Example 3.6, we know a minimal non-odd-transversal hypergraph can contain both even-sized edges and odd-sized edges. In the following we will discuss minimal non-odd-transversal hypergraphs only with even-sized edges.
Corollary 3.7. Let G be an minimal non-odd-transversal hypergraph only with even-sized edges, which has n vertices and m edges. Then the following results hold.
(3) The incidence bipartite graph Γ G has a matching M such that E(G) is matched by M , namely, there exists an injection f : E(G) → V (G) such that f (e) ∈ e for each e ∈ E(G).
Proof. Consider the incidence matrix B G of G. As G contains only even-sized edges, each row sum of B G is zero over Z 2 , which implies rankB G ≤ n − 1. By Theorem 3.4(3), rankB G = m − 1, yielding the result (1).  Corollary 3.9. Let G be a square hypergraph only with even-sized edges and evendegree vertices. Then G is minimal non-odd-transversal if and only if its dual G * is minimal non-odd-transversal.
Proof. Suppose G is minimal non-odd-transversal with n vertices (edges). By Corollary 3.8 (2), no two vertices of G lie in precisely the same edges of G. So G * is also square, and B G * = B ⊤ G . As each edge of G is even sized, each vertex of G * has even degree. So G * is minimal non-odd-transversal by Theorem 3.4. As G is isomorphic to (G * ) * , G is minimal non-odd-transversal if G * is.

minimal non-odd-bipartite regular hypergraphs
In this section we mainly discuss minimal non-odd-transversal k-uniform hypergraphs G. By the following lemma, k is necessarily even. So the minimal nonodd-transversal uniform hypergraphs are exactly the minimal non-odd-bipartite hypergraphs.
Lemma 4.1. Let G be a minimal non-odd-transversal k-uniform hypergraphs G, which has n vertices and m edges. Then k is even. If G is further d-regular, then d is even and d ≤ k.
Proof. By Theorem 3.4(4), each vertex of G has an even degree so that the sum of degrees of the vertices of G is even, which is equal to mk. As m is odd by Theorem 3.4, k is necessarily even, which implies that n ≥ m by Corollary 3.7(1). If G is d-regular, d is even by Theorem 3.4(4). Surely nd = mk, so d ≤ k as n ≥ m.
So, in the following discussion we only deal with non-odd-bipartite regular hypergraphs with even uniformity and even degree. 4.1. 2-regular minimal non-odd-bipartite hypergraphs. It is known that the only minimal non-bipartite simple graph is an odd cycle C 2l+1 , which is 2-regular.
As a simple generalization, the generalized power hypergraph C k, k 2 2l+1 is a 2-regular minimal non-odd-bipartite k-uniform hypergraph. However, the above hypergraph is not the only 2-regular minimal non-odd-bipartite k-uniform hypergraph. For example, the following 4-uniform hypergraph on 10 vertices with 5 edges below is minimal non-odd-bipartite:  Proof. Let H be a nonempty proper edge-induced sub-hypergraph of G. As G is connected, H contains a vertex v, which is also contained in some edge not in H.
So v has degree 1 in H. The result follows by Theorem 3.4(4).
Next we give a construction of 2-regular k-uniform hypergraphs, where k is an even integer greater than 2. Define a hypergraph G with vertex set [n], whose edges are So v contained in exactly two edges e s and e t , implying v has degree 2. Finally we note that G contains no multiple edges; otherwise, if e s = e t for s = t, then V s ∪ W s = V t ∪ W t , which implies that V s = W t and V t = W s as V s ∪ V t = ∅ and W s ∪ W t = ∅, a contradiction to the assumption. The result follows. ThenḠ is a k-uniform k-regular multi-hypergraph on n vertices. The incidence bipartite graph ΓḠ ofḠ is k-regular.
Let KḠ be a complete bipartite graph with two parts V (Ḡ) and E(Ḡ). Let KḠ be obtained from KḠ by deleting the edges between the vertices of V t := { k 2 (t − 1) + 1, . . . , k 2 t} and the vertices of . ThenKḠ is an (n − k 2 )-regular bipartite graph. Considering the k-regular bipartite graph ΓḠ, it contains a perfect matching M . By a possible relabeling of the vertices, we may assume that for t ∈ [m], Now deleting the edges between the vertices of V t and the vertices of E t from ΓḠ for t ∈ [m], we arrive at a k 2 -regular bipartite graph denoted byΓḠ, which is a subgraph ofKḠ. NowΓḠ, and henceKḠ has a perfect matchingM , where, for As G contains no multiple edges, if W t = V s for some s = t, surely W s = V t ; otherwise e t = e s = V t ∪ V s , a contradiction.
From the above discussion, KḠ andKḠ are respectively isomorphic to K n,n and K n,n . A perfect matchingM inKḠ is isomorphic to a perfect matching inK n,n . So G can be constructed as in Construction 4.3.

4.2.
Examples of d-regular minimal non-odd-bipartite hypergraphs. We first give an example of k-regular k-uniform minimal non-odd-bipartite hypergraph by using Cayley hypergraph. Let G = (Z n ; {1, 2, . . . , k−1}) be a Cayley hypergraph, where V (G) = Z n , and E(G) consists of edges {i, i + 1, . . . , i + k} for i ∈ Z n . Then G is connected, k-uniform and k-regular, with n vertices and n edges.
So x i = x i+k for each i ∈ Z n . Let t := gcd(k, n). Then there exist integers p, q such that pk + qn = t. Note that t is odd as n is odd, and if writing k = st, then s is even as k is even. For each i ∈ Z n , As s is even, for any x 1 , . . . , x t ∈ Z 2 , and any edge x 1 + · · · + x k = (x 1 + · · · + x t ) + · · · + (x (s−1)t+1 + · · · + x st ) = s(x 1 + · · · + x t ) = 0.
So, the solution space of B G x = 0 over Z 2 has dimension t, which implies that rankB G = n − t over Z 2 . The result now follows.
Let G be a k-uniform hypergraph with n vertices and m edges. Let G 1 , G 2 , . . . , G t be t disjoint copies of G. For each vertex v (or each edge e) of G, it has t copies v 1 , . . . , v t (or e 1 , . . . , e t ) in G 1 , . . . , G t respectively. Let t• G be a hypergraph whose vertex set is ∪ t i=1 V (G i ), and edge set is {e 1 ∪ · · · ∪ e t : e ∈ E(G)}. Then t • G is tk-uniform hypergraph with tn vertices and m edges, and the degree of v i in t • G is same as the degree of v in G for each v ∈ V (G) and i ∈ [t]. If further G is d-regular, then t • G is also d-regular.  Note that in Corollary 4.10, if d = 2, then H is an odd cycle C m , and t • C m = C 2t,t m (a generalized power hypergraph), both of which are minimal non-oddbipartite.
Thirdly we use a projective plane (X, B) of order q to construct a regular minimal non-odd-bipartite hypergraph. Recall a projective plane of order q consists of a set X of q 2 + q + 1 elements called points, and a set B of (q + 1)-subsets of X called lines, such that any two points lie on a unique line. It can be derived from the definition that any points lies on q + 1 lines, and two lines meet in a unique point, and there are q 2 + q + 1 lines. Now define a hypergraph based on (X, B), denoted by G = (X, B), whose vertices are the points of X and edges are the lines of B. Then G = (X, B) is a (q + 1)-regular (q + 1)-uniform hypergraph with q 2 + q + 1 vertices.
Theorem 4.11. Let (X, B) be a projective plane of order q, and let G = (X, B) be a hypergraph defined as in the above. If q is odd, then G = (X, B) is minimal non-odd-bipartite.
Proof. Let e be an edge of G = (X, B) or a line of (X, B). Then G−e = qI + J, where I is the identity matrix, and J is an all-ones matrix, both of size q 2 + q. So det B G−e B ⊤ G−e = det(qI + J) = (q 2 + 2q)q q 2 +q−1 ≡ 1 mod 2, implying that rankB G = m − 1 over Z 2 . The result follows by Theorem 3.4(3).
It is known that if q is an odd prime power, then there always exists a projective plane of order q by using the vector space F 3 q . By Lemma 4.9 and Theorem 4.11, we easily get the following result.
Remark 4.13. From Corollaries 4.8 and 4.12, the minimal non-odd-bipartite hypergraphs G have degree d and edge number m such that gcd(d, m) = 1. (Note that gcd(q + 1, q 2 + q + 1) = 1.) As gcd(d, m) = 1, from the equality nd = mk, we have d | k, where n, k are the number of vertices and the uniformity of G respectively.
In fact, there exist d-regular minimal non-odd-bipartite hypergraphs with m edges such that gcd(d, m) > 1. For example, let G be a 6-uniform 6-regular hypergraph with 9 edges below: By Theorem 3.4, it is also easy to verify that G is minimal non-odd-bipartite.
There also exist d-regular k-uniform minimal non-odd-bipartite hypergraphs such that d ∤ k. For example, let G be a 6-regular 8-uniform hypergraph with 9 edges below: By Theorem 3.4, it is easy to verify that G is minimal non-odd-bipartite.
Example 4.14. The minimal non-odd-bipartite uniform hypergraphs with fewest edges. By Theorem 3.4, if G is a k-uniform minimal non-odd-bipartite hypergraph with n vertices and m edges, then m is odd. If m = 1, G is surely odd-bipartite. So, m ≥ 3, and hence the maximum degree is at most 3 if m = 3. Assume that m = 3. By Theorem 3.4, each vertex has an even degree, implying that G is 2regular. So 2n = 3k, and 3|n. Letting n = 3l, we have k = 2l. So G = C 2l,l 3 , which is the unique example of minimal non-odd-bipartite hypergraph with 3 edges. It is consistent with the fact that C 3 is the unique minimal non-bipartite simple graph with 3 edges by taking k = 2.
Example 4.15. The minimal non-odd-bipartite uniform hypergraphs with fewest vertices. If G is a k-uniform minimal non-odd-bipartite hypergraph with n vertices and m edges. Then n ≥ k + 1, as an edge is odd-bipartite. Assume that n = k + 1. Then m ≤ k+1 k = k + 1, with equality if and only if G is a (k + 1)-simplex [7], i.e. any k vertices of G forms an edge. Let ∆ be the maximum degree of G, which is even by Theorem 3.4. As m is odd by Theorem 3.4, we have which implies that m = k + 1 and k is even. So, the (k + 1)-simplex is the unique example of k-uniform minimal non-odd-bipartite hypergraph with k + 1 vertices by Theorem 3.4. If taking k = 2, then C 3 is the the unique minimal non-bipartite simple graph with 3 vertices. It is easy to verify that G is non-regular minimal non-odd-bipartite by Theorem 3.4. where {1, 2, 10, 11} is a cut edge of G. By Theorem 3.4, G is minimal non-oddbipartite.

Least H-eigenvalue of minimal non-odd-bipartite hypergraphs
Let G be a k-uniform minimal hypergraph. Let x ∈ C V (G) whose entries are indexed by the vertices of G. For a subset U of V (G), denote x U := Π v∈U x u . Then we have Observe that there exists a vertex u such that x u ≤ 1 n 1/k . Letē be an edge of G containing u. Then By the definition, G −ē is odd-bipartite with an odd-bipartition {U, U c }. Now define a vector y on the vertices of G −ē such that y v = x v if v ∈ U and y v = −x v otherwise. Note thatē intersects U c in an even number of vertices as G is non-oddbipartite, which implies that yē = xē > 0. By Lemma 2.2(2) and Eq. (5.2), λ min (G) ≤ A(G)y k = −A(G)x k + 2kxē ≤ −ρ(G) + 2k n 1/k . For the second result, from Eq. (5.2), there exists one edgeê such that kxê is not greater than the average of the summands kx e over all m edges e of G, that is, Note that G −ê is also odd-bipartite with an odd-bipartition say {W, W c }. Now define a vector z on the vertices of G −ê such that z v = x v if v ∈ W and z v = −x v otherwise. By a similar discussion as the above, we have Corollary 5.2. Let k be a positive even integer. For any ǫ > 0, for any k-uniform minimal non-odd-bipartite hypergraph G with sufficiently larger number of vertices or edges, (1) −ρ(G) < λ min (G) < −ρ(G) + ǫ, (2) −1 < λ min (G)/ρ(G) < −1 + ǫ.
For a connected k-uniform hypergraph G, where k is even, if we denote α(G) := ρ(G) + λ min (G), β(G) := −λ min (G)/ρ(G), then by Lemma 2.3, α(G) ≥ 0, with equality if G is odd-bipartite; and 0 < β(G) ≤ 1, with right equality if and only if G is odd-bipartite. So we can use α(G) and β(G) to measure the non-odd-bipartiteness of an even uniform hypergraph. Furthermore, by Theorem 5.1 and Corollary 5.2, if G is minimal non-oddbipartite hypergraph, then α(G) → 0 and β(G) → 1 when the number of vertices or edges of G goes to infinity. So, the minimal non-odd-bipartite hypergraphs are very close to be odd-bipartite in this sense.