A note on improved upper bounds on the transversal number of hypergraphs

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

On Upper Transversals in 3-Uniform Hypergraphs

A set $S$ of vertices in a hypergraph $H$ is a transversal if it has a nonempty intersection with every edge of $H$. The upper transversal number $\Upsilon(H)$ of $H$ is the maximum cardinality of a minimal transversal in $H$. We show that if $H$ is a connected $3$-uniform hypergraph of order $n$, then $\Upsilon(H) > 1.4855 \sqrt[3]{n} - 2$. For $n$ sufficiently large, we construct infinitely many connected $3$-uniform hypergraphs, $H$, of order~$n$ satisfying $\Upsilon(H) < 2.5199 \sqrt[3]{n}$. We conjecture that $\displaystyle{\sup_{n \to \infty} \, \left( \inf \frac{ \Upsilon(H) }{ \sqrt[3]{n} } \right) = \sqrt[3]{16} }$, where the infimum is taken over all connected $3$-uniform hypergraphs $H$ of order $n$.

Transversals and Independence in Linear Hypergraphs with Maximum Degree Two

For $k \ge 2$, let $H$ be a $k$-uniform hypergraph on $n$ vertices and $m$ edges. Let $S$ be a set of vertices in a hypergraph $H$. The set $S$ is a transversal if $S$ intersects every edge of $H$, while the set $S$ is strongly independent if no two vertices in $S$ belong to a common edge. The transversal number, $\tau(H)$, of $H$ is the minimum cardinality of a transversal in $H$, and the strong independence number of $H$, $\alpha(H)$, is the maximum cardinality of a strongly independent set in $H$. The hypergraph $H$ is linear if every two distinct edges of $H$ intersect in at most one vertex. Let $\mathcal{H}_k$ be the class of all connected, linear, $k$-uniform hypergraphs with maximum degree $2$. It is known [European J. Combin. 36 (2014), 231–236] that if $H \in \mathcal{H}_k$, then $(k+1)\tau(H) \le n+m$, and there are only two hypergraphs that achieve equality in the bound. In this paper, we prove a much more powerful result, and establish tight upper bounds on $\tau(H)$ and tight lower bounds on $\alpha(H)$ that are achieved for infinite families of hypergraphs. More precisely, if $k \ge 3$ is odd and $H \in \mathcal{H}_k$ has $n$ vertices and $m$ edges, then we prove that $k(k^2 - 3)\tau(H) \le (k-2)(k+1)n + (k - 1)^2m + k-1$ and $k(k^2 - 3)\alpha(H) \ge (k^2 + k - 4)n - (k-1)^2 m - (k-1)$. Similar bounds are proven in the case when $k \ge 2$ is even.

Transversals in 4-Uniform Hypergraphs

Let $H$ be a $4$-uniform hypergraph on $n$ vertices. The transversal number $\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. The result in [J. Combin. Theory Ser. B 50 (1990), 129—133] by Lai and Chang implies that $\tau(H) \le 7n/18$ when $H$ is $3$-regular. The main result in [Combinatorica 27 (2007), 473—487] by Thomassé and Yeo implies an improved bound of $\tau(H) \le 8n/21$. We provide a further improvement and prove that $\tau(H) \le 3n/8$, which is best possible due to a hypergraph of order eight. More generally, we show that if $H$ is a $4$-uniform hypergraph on $n$ vertices and $m$ edges with maximum degree $\Delta(H) \le 3$, then $\tau(H) \le n/4 + m/6$, which proves a known conjecture. We show that an easy corollary of our main result is that if $H$ is a $4$-uniform hypergraph with $n$ vertices and $n$ edges, then $\tau(H) \le \frac{3}{7}n$, which was the main result of the Thomassé-Yeo paper [Combinatorica 27 (2007), 473—487].

X− Dominating colour transversals in graphs

Let G = (X, Y,E) be a bipartite graph. A X-dominating set D ⊆X is called a X−dominating colour transversal set of a graph G if D isa transversal of at least one $chi$−partition of G.The minimum cardinal-ity of a X−dominating colour transversal set is called X−dominatingcolour transversal number and is denoted by $chi_{dct}(G)$. We find thebounds of X−dominating colour transversal number and characterizethe graphs attaining the bound.