Locating-total dominating sets in twin-free graphs: a conjecture

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of $G$, denoted $LT(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \geq 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $LT(G) \leq \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $LT(G) \le \frac{3}{4}n$.Comment: 18 pages, 1 figur

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

Location-domination in line graphs

A set $D$ of vertices of a graph $G$ is locating if every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$, where $N(u)$ denotes the open neighborhood of $u$. If $D$ is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of $G$. A graph $G$ is twin-free if every two distinct vertices of $G$ have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning. Locating-total dominating sets in twin-free graphs: a conjecture. The Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any twin-free graph $G$ without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.Comment: 23 pages, 2 figure

A Greedy Partition Lemma for Directed Domination

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $\gamma(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $\Gamma_d(G)$, which is the maximum directed domination number $\gamma(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if $\alpha$ denotes the independence number of a graph $G$, we show that $\alpha \le \Gamma_d(G) \le \alpha(1+2\ln(n/\alpha))$.Comment: 12 page