Bounds on the Game Transversal Number in Hypergraphs

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).Comment: 23 pages

Regularity of squarefree monomial ideals

We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.

Transversals in 44-Uniform Hypergraphs

Let $H$ be a $3$-regular $4$-uniform hypergraph on $n$ vertices. The transversal number $\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990), 129--133] proved that $\tau(H) \le 7n/18$. Thomass\'{e} and Yeo [Combinatorica 27 (2007), 473--487] improved this bound and showed that $\tau(H) \le 8n/21$. 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 the total domination number of a graph on $n$ vertices with minimum degree at least~4 is at most $3n/7$, which was the main result of the Thomass\'{e}-Yeo paper [Combinatorica 27 (2007), 473--487].Comment: 41 page

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196