Bounds on the Game Transversal Number in Hypergraphs

TL;DR

This paper investigates the game transversal number in hypergraphs, establishing upper bounds for 3-uniform and 4-uniform hypergraphs and comparing it with the classical transversal number, advancing understanding of combinatorial game parameters.

Contribution

It introduces new bounds for the game transversal number in uniform hypergraphs, extending previous results and analyzing its relation to the transversal number.

Findings

For 3-uniform hypergraphs, /16 of ( H + mH) bounds /16 au_g(H)

For 4-uniform hypergraphs, 71/252 of ( H + mH) bounds 1/252 au_g(H)

Established the Transversal Continuation Principle and compared /16 and 1/252 bounds with known bounds

Abstract

Let $H = (V,E)$ be a hypergraph with vertex set $V$ and edge set $E$ of order $\nH = |V|$ and size $\mH = |E|$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. A vertex hits an edge if it belongs to that edge. The transversal game played on $H$ involves of two players, \emph{Edge-hitter} and \emph{Staller}, who take turns choosing a vertex from $H$. Each vertex chosen must hit at least one edge not hit by the vertices previously chosen. The game ends when the set of vertices chosen becomes a transversal in $H$. Edge-hitter wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The \emph{game transversal number}, $\tau_g(H)$, of $H$ is the number of vertices chosen when Edge-hitter starts the game and both players play optimally. We compare the game transversal number of a hypergraph with its transversal number, and also present an important fact concerning the monotonicity of $\tau_g$, that we call the Transversal Continuation Principle. It is known that if $H$ is a hypergraph with all edges of size at least~$2$, and $H$ is not a $4$-cycle, then $\tau_g(H) \le \frac{4}{11}(\nH+\mH)$; and if $H$ is a (loopless) graph, then $\tau_g(H) \le \frac{1}{3}(\nH + \mH + 1)$. We prove that if $H$ is a $3$-uniform hypergraph, then $\tau_g(H) \le \frac{5}{16}(\nH + \mH)$, and if $H$ is $4$-uniform, then $\tau_g(H) \le \frac{71}{252}(\nH + \mH)$.