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A graph is (I , F) - partitionable if its vertex set can be partitioned into two parts such that one part I is an independent set, and the other F induces a forest. A graph is k - degenerate if every subgraph H contains a vertex of degree at most k in H. Bernshteyn and Lee defined a generalization of k -degenerate graphs, which is called weakly k -degenerate. In this paper, we show that planar graphs without 4-, 7-, 9-cycles, and 5-cycles normally adjacent to 3-cycles are both (I , F) -partitionable and weakly 2-degenerate. [ABSTRACT FROM AUTHOR]

It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection T = (T 1 , ⋯ , T m) of not-necessarily distinct tournaments on a common vertex set V, an m-edge directed graph D with vertices in V is called a T -transversal if there exists a bijection ϕ : E (D) → [ m ] such that e ∈ E (T ϕ (e)) for all e ∈ E (D) . We prove that for sufficiently large m with m = | V | - 1 , there exists a T -transversal Hamilton path. Moreover, if m = | V | and at least m - 1 of the tournaments T 1 , ... , T m are assumed to be strongly connected, then there is a T -transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub H -partition. [ABSTRACT FROM AUTHOR]

We prove that for all n>1 $n\gt 1$ every latin n $n$‐dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer‐latin cubes of order 5 with no transversals. For each n≥3 $n\ge 3$ and q≥3 $q\ge 3$ we construct a (2q−2)×q×⋯×q $(2q-2)\times q\times {\rm{\cdots }}\times q$ latin n $n$‐dimensional cuboid of order q $q$ with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5. [ABSTRACT FROM AUTHOR]

In this work subultra-groups of an ultra-group of a subgroup over a finite group are studied. Also, a subultra-group generated by a subset of an ultra-group is worked and then all the elements are determined in terms of the subset. Moreover, in which conditions the intersection of two normal subultra-groups is normal subultra-group is given. Quotient ultra-groups are also studied. [ABSTRACT FROM AUTHOR]

The transversal number τ(H) $\tau (H)$ of a hypergraph H $H$ is the minimum number of vertices that intersect every edge of H $H$. This notion of transversal is fundamental in hypergraph theory and has been studied a great deal in the literature. A hypergraph H $H$ is r $r$‐regular if every vertex of H $H$ has degree r $r$, that is, every vertex of H $H$ belongs to exactly r $r$ edges. Further, H $H$ is k $k$‐uniform if every edge of H $H$ has size k $k$, and so every edge of H $H$ is a k $k$‐element subset of V(H) $V(H)$. For r≥2 $r\ge 2$ and k≥2 $k\ge 2$, let Hr,k,nreg ${{\rm{ {\mathcal H} }}}_{r,k,n}^{\text{reg}}$ be the class of all r $r$‐regular k $k$‐uniform hypergraphs of order n $n$. In this paper we study the problem posed by Tuza to determine or estimate the best possible constants θr,k ${\theta }_{r,k}$ (which depend only on r $r$ and k $k$) for each r≥2 $r\ge 2$ and k≥2 $k\ge 2$, such that τ(H)≤θr,k×n $\tau (H)\le {\theta }_{r,k}\times n$ for all H∈Hr,k,nreg $H\in {{\rm{ {\mathcal H} }}}_{r,k,n}^{\text{reg}}$. These constants are given by θr,k=supτ(H)n ${\theta }_{r,k}=\text{sup}\frac{\tau (H)}{n}$, where the supremum is taken over all H∈Hr,k,nreg $H\in {{\rm{ {\mathcal H} }}}_{r,k,n}^{\text{reg}}$. Tuza presented closed formulas when r=2 $r=2$ or k=2 $k=2$, and showed that θr,2=rr+1 ${\theta }_{r,2}=\frac{r}{r+1}$ for all r≥2 $r\ge 2$, θ2,k=43k ${\theta }_{2,k}=\frac{4}{3k}$ for k≥2 $k\ge 2$ even, and θ2,k=43k+1 ${\theta }_{2,k}=\frac{4}{3k+1}$ for k≥3 $k\ge 3$ odd. We conjecture that θ3,k≤32k ${\theta }_{3,k}\le \frac{3}{2k}$ for all k≥2 $k\ge 2$ and that θ4,k≤127k ${\theta }_{4,k}\le \frac{12}{7k}$ for all k≥2 $k\ge 2$. We show that both these conjectures hold for k∈{2,3,4,6} $k\in \{2,3,4,6\}$. Moreover we show, for example, that θ3,3=12 ${\theta }_{3,3}=\frac{1}{2}$ and θ4,3=59 ${\theta }_{4,3}=\frac{5}{9}$. We show that for every 0<ε<1 $0\lt \varepsilon \lt 1$ and for sufficiently large r $r$ and k $k$, the growth of θr,k ${\theta }_{r,k}$ is given by ε1+ln(r)k≤θr,k≤1+ln(r)k $\varepsilon \left(\frac{1+\mathrm{ln}(r)}{k}\right)\le {\theta }_{r,k}\le \frac{1+\mathrm{ln}(r)}{k}$. [ABSTRACT FROM AUTHOR]

We study the (a, b)-monochromatic transversal game that is a combinatorial Maker–Breaker game where Alice and Bob alternately colour a vertices in red and b vertices in blue of a hypergraph, respectively. Either player is enabled to start the game. Alice tries to construct a hyperedge transversal, and Bob tries to prevent this. The winner is Alice if she obtains a red hyperedge transversal; otherwise, Bob wins the game if he obtains a monochromatic blue hyperedge. Maker–Breaker games were determined to be PSPACE-complete. In this work, we analyze the game played on clique-hypergraphs of powers of cycles, and we show strategies that, depending on the choice of the parameters, allow a specific player to win the game. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to prove the following result: Let be a -disjoint finite family of disks with the property that every four of them have a common line transversal. Then there exists a line transversal to all members of . Moreover, we prove that the number is sharp. [ABSTRACT FROM AUTHOR]

The (a , b) - monochromatic transversal game is a combinatorial Maker-Breaker game where two players, Alice and Bob, alternately take turns colouring a and b vertices of a hypergraph in red and blue, respectively (either player may start the game). Alice tries to construct a hyperedge transversal, and Bob tries to prevent this. Alice wins if she obtains a red hyperedge transversal, and Bob wins if he obtains a monochromatic blue hyperedge. We study the game played on biclique-hypergraphs of powers of paths and powers of cycles, showing strategies that, depending on the choice of parameters a and b , allow a specific player to win the game. [ABSTRACT FROM AUTHOR]

The recursive differentiability of finite binary quasigroups is investigated. We consider the Bruck and Belousov constructions of prolongation of finite quasigroups and give necessary and sufficient conditions when such prolongations are recursively 1-differentiable. [ABSTRACT FROM AUTHOR]

A biclique of a graph G is a maximal complete bipartite induced subgraph of G with at least one edge. We define and study the time complexity of the problems of finding the minimum biclique transversal and maximum biclique independent set of a graph G, denoted by α b (G) and τ b (G) respectively. We prove that the Biclique-Transversal and Biclique-Independent-Set problems are NP-Complete for the classes of split graphs, planar graphs C 4 -free with ∆ = 4, and bipartite C 4 -free graphs with ∆ = 4. In addition, we provide polinomial time algorithms for block graphs and split gem-free graphs. Finally, we introduce the concept of biclique-perfectness. [ABSTRACT FROM AUTHOR]

The transversal number Τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard [Discuss. Math. Graph Theory, 22 (2002), pp. 199-210] conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then Τ(H) < 1/4n. In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture. [ABSTRACT FROM AUTHOR]

This is the second instalment of a two-part thematic review of nineteenth-century French research in the Anglophone sphere. The review sets aside the standard sub-disciplinary categories in order to envision a laboratorium, a space of multiple confluent and contiguous lines of exploratory work. In this space, more fluid thematic connections surface between projects irrespective of their focus, their methodology, or their scale, revealing some of the major transversal routes of research across the recent past and the present, and, speculatively, into the future. The two-part review explores biography / body / earth / emotion / experiment / movement / thing. [ABSTRACT FROM AUTHOR]

A transversal in a hypergraph H is set of vertices that intersect every edge of H. A transversal coalition in H consists of two disjoint sets of vertices X and Y of H , neither of which is a transversal but whose union X ∪ Y is a transversal in H. Such sets X and Y are said to form a transversal coalition. A transversal coalition partition in H is a vertex partition Ψ = { V 1 , V 2 , ... , V p } such that for all i ∈ [ p ] , either the set V i is a singleton set that is a transversal in H or the set V i forms a transversal coalition with another set V j for some j , where j ∈ [ p ] ∖ { i }. The transversal coalition number C τ (H) in H equals the maximum order of a transversal coalition partition in H. For k ≥ 2 a hypergraph H is k -uniform if every edge of H has cardinality k. Among other results, we prove that if k ≥ 2 and H is a k -uniform hypergraph, then C τ (H) ≤ ⌊ 1 4 k 2 ⌋ + k + 1. Further we show that for every k ≥ 2 , there exists a k -uniform hypergraph that achieves equality in this upper bound. [ABSTRACT FROM AUTHOR]

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Muslim women who are forcibly displaced from their homophobic parent culture are further oppressed within the structures of Islamophobia in the host culture. Discursive re-imaginations based on their Islamic authenticity have skewed them as either veiled or unveiled women from other homogeneous ethnic cultures. Such cursory representations have cemented perceptions about them as outsiders and so a source of constant threat. Therefore, the main objective of this article is to aid discursive reconfigurations of partial representations affecting transculturally or forcibly displaced Muslim women at the intersections of racial, gender or religious persecutions. Through a reading of the life-narratives by similarly displaced Muslim women in the anthology It's Not About the Burqa, this article examines two key questions: 1) How do these women negate discursive conceptions of single stories? 2) How do they reach a unique discourse that addresses such partial representations? The article proposes heterotopia and hypomnemata as two transversal possibilities. Heterotopias are worlds within worlds which mirror what is outside. They can contain differences and undesirable bodies. Whereas hypomnemata refers to personal notes used for later reading and meditation. There has been little work bringing together the transcultural and the transversal. While the transcultural is a system of thought that conceives of cultures as an ongoing flux of confluences, transversality endorses plural possibilities. It offers tools to deterritorialize closed logics.2 This article therefore calls for revisiting the collaborative dynamics of these concepts as contested through the life-narratives of displaced Muslim women. [ABSTRACT FROM AUTHOR]

An idempotent Latin square of order v $v$ is called resolvable and denoted by RILS(v) if the v(v−1) $v(v-1)$ off‐diagonal cells can be resolved into v−1 $v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v $v$, briefly LRILS(v), is a collection of v−2 $v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v∈{14,20,22,28,34,35,38,40,42,46,50,55,62} $v\in \{14,20,22,28,34,35,38,40,42,46,50,55,62\}$ by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer v≥3 $v\ge 3$, except v=6 $v=6$. [ABSTRACT FROM AUTHOR]

Let H be a hypergraph of order n H = | V (H) | and size m H = | E (H) |. The transversal number τ (H) of H is the minimum number of vertices that intersect every edge of H. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A k -uniform hypergraph has all edges of size k. Let L k be the class of k -uniform linear hypergraphs. In this paper we study the problem of determining or estimating the best possible constants q k (which depends only on k) such that τ (H) ≤ q k (n H + m H) for all H ∈ L k . We establish lower bounds on q k for all k ≥ 2. For all k ≥ 2 and for all 0 < a < 1 , we show that q k ≥ a (1 − a) k / ( (1 − a) k + a ln (e / a)) , where here ln denotes the natural logarithm to the base e. [ABSTRACT FROM AUTHOR]

Let lct(G) be the minimum cardinality of a set of vertices that intersects every longest cycle of a 2‐connected graph G. We show that lct(G)≤k−1 if G is a partial k‐tree and that lct(G)≤max{1,ω(G)−3} if G is chordal, where ω(G) is the cardinality of a maximum clique in G. Those results imply that all longest cycles intersect in 2‐connected series‐parallel graphs and in 3‐trees. [ABSTRACT FROM AUTHOR]

The (a,b)-monochromatic transversal game is an avoider-enforcer combinatorial game in which two players, Alice and Bob, alternately take turns colouring respectively a vertices in red and b vertices in blue of a hypergraph. She wins the game by obtaining a red transversal while he wins by obtaining a monochromatic blue hyperedge. Also, both players are enabled to start the game and they play optimally. In this paper, we analyze the game played on biclique-hypergraphs of powers of paths and of powers of cycles showing strategies that, depending on the choice of the parameters a and b, allow a specific player to win the game. [ABSTRACT FROM AUTHOR]

This is the first instalment of a two-part thematic review of nineteenth-century French research in the Anglophone sphere. The review sets aside the standard sub-disciplinary categories in order to envision a laboratorium, a space of multiple confluent and contiguous lines of exploratory work. In this space, more fluid thematic connections surface between projects irrespective of their focus, their methodology, or their scale, revealing some of the major transversal routes of research across the recent past and the present, and, speculatively, into the future. The two-part review explores biography / body / earth / emotion / experiment / movement / thing. [ABSTRACT FROM AUTHOR]

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit longest path transversals of constant size. The same technique allows us to show that 2-connected graphs admit sublinear longest cycle transversals. [ABSTRACT FROM AUTHOR]

A subset T of vertices in a hypergraph H is a transversal if T has a nonempty intersection with every edge of H. The transversal number τ (H) of H is the minimum size of a transversal in H. A hypergraph H is 3-uniform if every edge of H has size 3. Let H be a 3-uniform hypergraph with n H vertices and m H edges. Tuza (Discrete Math 86:117–126, 1990) and Chvátal and McDiarmid (Combinatorica 12:19–26, 1992) showed that 4 τ (H) ≤ n H + m H . Chvátal and McDiarmid also showed that 6 τ (H) ≤ 2 n H + m H . The linear hypergraphs achieving equality in these bounds were characterized by the authors (Henning and Yeo in J Graph Theory 59:326–348, 2008; Discrete Math 313:959–966, 2013). In this paper, we show that these bounds can be improved if we impose some structural properties on H. We show that if H does not contain a subhypergraph isomorphic to the affine plane AG(2, 3) of order 3 with two vertices deleted, then 17 τ (H) ≤ 5 n H + 3 m H . The total domination number γ t (G) of a graph G is the minimum cardinality of a set S of vertices so that every vertex in G is adjacent to some vertex in S. It is known (Archdeacon et al. in J Graph Theory 46:207–210, 2004) that if G is a graph of order n with minimum degree at least 3, then γ t (G) ≤ 1 2 n , and that this bound is tight. As a consequence of our hypergraph results, we show that if G is a graph of order n with minimum degree at least 3 that contains no 4-cycles and no specified graph on 12 vertices as a subgraph, then γ t (G) ≤ 8 17 n . [ABSTRACT FROM AUTHOR]

As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let T λ (τ) and ST λ (τ) denote the set of τ -avoiding transversals and τ -avoiding symmetric transversals of a Young diagram λ , respectively. In this paper, we are mainly concerned with the distribution of the peak set and the valley set on standard Young tableaux and pattern avoiding transversals. In particular, we prove that the peak set and the valley set are equidistributed on the standard Young tableaux of shape λ / μ for any skew diagram λ / μ. The equidistribution enables us to show that the peak set is equidistributed over T λ (12 ⋯ k τ) (resp. ST λ (12 ⋯ k τ)) and T λ (k ⋯ 21 τ) (resp. ST λ (k ⋯ 21 τ)) for any Young diagram λ and any permutation τ of { k + 1 , k + 2 , ... , k + m } with k , m ≥ 1. Our results are refinements of the result of Backelin-West-Xin which states that | T λ (12 ⋯ k τ) | = | T λ (k ⋯ 21 τ) | and the result of Bousquet-Mélou and Steingrímsson which states that | ST λ (12 ⋯ k τ) | = | ST λ (k ⋯ 21 τ) |. As applications, we are able to • confirm a recent conjecture posed by Yan-Wang-Zhou which asserts that the peak set is equidistributed over 12 ⋯ k τ -avoiding involutions and k ⋯ 21 τ -avoiding involutions; • prove that alternating involutions avoiding the pattern 12 ⋯ k τ are equinumerous with alternating involutions avoiding the pattern k ⋯ 21 τ , paralleling the result of Backelin-West-Xin for permutations, the result of Bousquet-Mélou and Steingrímsson for involutions, and the result of Yan for alternating permutations. [ABSTRACT FROM AUTHOR]

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is n + a if and only if the maximum size of a partial transversal is either n - 2 a or n - 2 a + 1 . (2) A minimal cover in a Latin square of order n has size at most μ n = 3 (n + 1 / 2 - n + 1 / 4) . (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to μ n . (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to μ n . (5) If 1 ⩽ k ⩽ n / 2 and n ⩾ 5 then there is a Latin square of order n with a maximal partial transversal of size n - k . (6) For any ε > 0 , asymptotically almost all Latin squares have no maximal partial transversal of size less than n - n 2 / 3 + ε . [ABSTRACT FROM AUTHOR]

Two simple alternative methods of construction of optimal partial triallel cross designs (unblocked and blocked) for p > 3 lines are proposed by using transversal and parallel transversals of a Latin square of order p, where p is a prime or power of a prime. Sharma et al. (2011) and Sharma and Fanta (2012) also constructed these designs by using two mutually orthogonal Latin squares, but we constructed these designs only taking one Latin square. Therefore, our methods are simple in comparison to their methods. These designs are found to be optimal in the sense of Kiefer (1975) and Das and Gupta (1997). [ABSTRACT FROM AUTHOR]

Paul Erdős and László Lovász established by means of an example that there exists a maximal intersecting family of k -sets with ⌊ ( e − 1 ) k ! ⌋ blocks, where e is the base of natural logarithm. László Lovász conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this note we compute the transversal size of a certain series of intersecting families of k -sets, which is constructed over the Cyclic graph. It helps to provide an example of maximal intersecting family of k -sets with so many blocks and to present two worthwhile examples which disprove two special cases of one of the conjectures of Frankl et al. [ABSTRACT FROM AUTHOR]

Let Π q be an arbitrary finite projective plane of order q . A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S . Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to ⌈ 3 q ln q ⌉ + ⌈ ( q + 1 ) ∕ 2 ⌉ . The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs. [ABSTRACT FROM AUTHOR]

An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an [ABSTRACT FROM AUTHOR]

An n -ary quasigroup f of order q is an n -ary operation over a set of cardinality q such that the Cayley table of the operation is an n -dimensional latin hypercube of order q . A transversal in a quasigroup f (or in the corresponding latin hypercube) is a collection of q ( n + 1 ) -tuples from the Cayley table of f , each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011. An n -ary quasigroup f is called reducible if it can be obtained as a composition of two quasigroups whose arity is at least 2, and it is completely reducible if it can be decomposed into binary quasigroups. In this paper we investigate transversals in reducible quasigroups and in quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible quasigroups. Next we prove that, except for the iterated group Z 4 of even arity, every n -ary quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in quasigroups of order 4 and odd arity and count transversals in the iterated group Z 4 of odd arity and in the iterated group Z 2 2 . All results of this paper can be regarded as those concerning latin hypercubes. [ABSTRACT FROM AUTHOR]

Denote by A p k the Latin square of order n = p k formed by the Cayley table of the additive group ( Z p k , + ) , where p is an odd prime and k is a positive integer. It is shown that for each p there exists Q > 0 such that for all sufficiently large k , the number of transversals in A p k exceeds ( n Q ) n p ( p − 1 ) . [ABSTRACT FROM AUTHOR]

A transversal in a hypergraph H is a subset of vertices that has a nonempty intersection with every edge of H . A transversal family F of H is a family of (not necessarily distinct) transversals of H . The effective transversal-ratio of the family F is the ratio of the number of sets in F over the maximum times r F any element appears in F . The fractional disjoint transversal number FDT ( H ) is the supremum of the effective transversal-ratio taken over all transversal families. That is, FDT ( H ) = sup F | F | ∕ r F . Using a connection with not-all-equal 3 -SAT, we prove that if H is a 3 -regular 3 -uniform hypergraph, then FDT ( H ) ≥ 2 , which proves a known conjecture. Using probabilistic arguments, we prove that for all k ≥ 3 , if H is a k -regular k -uniform hypergraph, then FDT ( H ) ≥ 1 ∕ ( 1 − ( k − 1 k ) 1 k 1 k − 1 ) , and that this bound is essentially best possible. [ABSTRACT FROM AUTHOR]

We present a survey on transversal, T 1 -independent, and T 1 -complementary topologies on sets and groups. First we consider the complementarity problem in the respective lattices L ( X ) and L 1 ( X ) of all topologies and T 1 -topologies on an infinite set X . The second part of the article deals with the complementarity problem in the family G 2 ( G ) of Hausdorff topological group topologies on a given infinite group G . It turns out that NO element of G 2 ( G ) has a complement in G 2 ( G ) , so the main attention is paid to the study of transversality and T 1 -independence in G 2 ( G ) . The article contains well-known old results, recent results dispersed in many sources, and several new facts obtained by the authors of this contribution. Selected open problems are included here to indicate that this area of research is far from complete. [ABSTRACT FROM AUTHOR]

Let r ( k ) denote the maximum number of edges in a k -uniform intersecting family with covering number k . Erdős and Lovász proved that ⌊ k ! ( e − 1 ) ⌋ ≤ r ( k ) ≤ k k . Frankl, Ota, and Tokushige improved the lower bound to r ( k ) ≥ ( k / 2 ) k − 1 , and Tuza improved the upper bound to r ( k ) ≤ ( 1 − e − 1 + o ( 1 ) ) k k . We establish that r ( k ) ≤ ( 1 + o ( 1 ) ) k k − 1 . [ABSTRACT FROM AUTHOR]

The transversal number τ (H) of a hypergraphHis the minimum cardinality of a set of vertices that intersects all edges ofH. The matching number α′(H) ofHis the maximum cardinality of a matching inH. A hypergraphHis intersecting if and only if α′(H) = 1. For an intersecting hypergraphHof rankrwithout isolated vertex, clearly τ (H) ≤r.His called 1-special if τ (H) =r, andHis maximal 1-special ifHis 1-special and adding any missingr-edge toHincreases the matching number.n2(r) andn3(r) are the maximum orders of a 1-special hypergraph and a maximal 1-special hypergraph, respectively. Furthermore, the intersecting hyper-graphHis called 1-edge-critical if for anye∈E(H) and anyv∈e,v-shrinkingeincreases the matching number;His called 1-vertex-critical if for anyv∈V(H) deleting the vertexvandv-shrinking all edges incident withvincreases the matching number.n4(r) andn5(r) are the maximum orders of a 1-edge-critical hypergraph and a 1-vertex-critical hypergraph, respectively. The intersecting hypergraphs, as defined above, are said to be matching critical [M.A. Henning and A. Yeo, Quaest. Math.37(2014), 127–138]. In this paper we study the extremal behavior of matching critical intersecting hypergraphs. We show thatn2(r) =n3(r) andn4(r) =n5(r) for allr≥ 2, which partly answers an open problem on matching critical intersecting hypergraphs posed by Henning and Yeo. We also give a strengthening of the resultn4(r) =n5(r) for intersectingr-uniform hypergraphs. [ABSTRACT FROM AUTHOR]

The 3/4 -Game Total Domination Conjecture posed by Henning, Klavžar, and Rall [Combinatorica, (2016)] states that if G is a graph on n vertices in which every component contains at least three vertices, then γtg(G) ≤ 3/4n, where γtg(G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, τg(H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H ... C4, then τg(H) ≤ 4/11 (nH + mH), where nH and mH denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γtg(G) < 8 11n. As a consequence of this result, the 3/4 -Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2. [ABSTRACT FROM AUTHOR]

In [9] Thomassé and Yeo pose the following problem: Find the minimumckfor which everyk-uniform hypergraph withnvertices andnedges has a transversal of size at mostckn. A direct consequence of this result is that every graph of ordernwith minimum degree at leastkhas a total dominating set of cardinality at mostckn. It is known thatc2= 2/3,c3= 1/2, andc4= 3/7. Thomassé and Yeo show that 4/11 ≤c5and conjecture thatc5= 4/11. In this paper we show thatc5≤ 17/44. Thus, 16/44 ≤c5≤ 17/44. More generally we prove that every 5-uniform hypergraph onnvertices andmedges has a transversal with no more than (10n+7m)/44 vertices. Consequently, every graph onnvertices with minimum degree at least five has total domination number at most 17n/44. [ABSTRACT FROM PUBLISHER]

By the assistance of an independence space on a set S with finite rank, this paper presents a way to determine a family of subsets of S to possess an independent transversal. Afterwards, it provides another result on infinite transversal relative to partial transversals of a family of subsets of S. The two results are in the field of transversal theory. [ABSTRACT FROM AUTHOR]

The logarithm of the maximum number of transversals over all latin squares of order n is greater than n 6 ( ln n + O ( 1 ) ) . [ABSTRACT FROM AUTHOR]

We discuss the class of paratopological groups which admits a transversal, T 1 -independent and T 1 -complementary paratopological group topology. We show that the Sorgenfrey line does not admit a T 1 -complementary Hausdorff paratopological group topology, which gives a negative answer to [5, Problem 10]. We give a very useful criterion for transversality in term of submaximal paratopological group topology, and prove that if a non-discrete paratopological group topology G contains a central subgroup which admits a transversal paratopological group topology, then so does G. We introduce the concept of PT -sequence and give a characterization of an Abelian paratopological group being determined by a PT -sequence. As the applications, we prove that the Abelian paratopological group, which is endowed with the strongest paratopological group topology being determined by a T -sequence, does not admit a T 1 -complementary Hausdorff paratopological group topology on G. Finally, we study the class of countable paratopological groups which is determined by a PT -filter, and obtain a sufficient condition for a countable paratopological group G being determined by a PT -sequence which admits a transversal paratopological group topology on G being determined by a PT -sequence. [ABSTRACT FROM AUTHOR]

Let lpt (G) be the minimum cardinality of a transversal of longest paths in G , that is, a set of vertices that intersects all longest paths in a graph G. There are several results in the literature bounding the value of lpt (G) in general or in specific classes of graphs. For instance, lpt (G) = 1 if G is a connected partial 2-tree, and a connected partial 3-tree G is known with lpt (G) = 2. We prove that lpt (G) ≤ 3 for every connected partial 3-tree G ; that lpt (G) ≤ 2 for every planar 3-tree G ; and that lpt (G) = 1 if G is a connected bipartite permutation graph or a connected full substar graph. Our first two results can be adapted for broader classes, improving slightly some known general results: we prove that lpt (G) ≤ k for every connected partial k -tree G and that lpt (G) ≤ max { 1 , ω (G) − 2 } for every connected chordal graph G , where ω (G) is the cardinality of a maximum clique in G. [ABSTRACT FROM AUTHOR]