Distinguishing-Transversal in Hypergraphs and Identifying Open Codes in Cubic Graphs

Abstract

The open neighborhood N(v) of a vertex v in a graph G is the set of vertices adjacent to v in G. A graph is twin-free (or open identifiable) if every two distinct vertices have distinct open neighborhoods. A separating open code in G is a set C of vertices such that ${N(u) \cap C \neq N(v) \cap C}$ for all distinct vertices u and v in G. An open dominating set, or total dominating set, in G is a set C of vertices such that ${N(u) \cap C \ne N(v) \cap C}$ for all vertices v in G. An identifying open code of G is a set C that is both a separating open code and an open dominating set. A graph has an identifying open code if and only if it is twin-free. If G is twin-free, we denote by ${\gamma^{\rm IOC}(G)}$ the minimum cardinality of an identifying open code in G. A hypergraph H is identifiable if every two edges in H are distinct. A distinguishing-transversal T in an identifiable hypergraph H is a subset T of vertices in H that has a nonempty intersection with every edge of H (that is, T is a transversal in H) such that T distinguishes the edges, that is, ${e \cap T \neq f \cap T}$ for every two distinct edges e and f in H. The distinguishing-transversal number ${\tau_D(H)}$ of H is the minimum size of a distinguishing-transversal in H. We show that if H is a 3-uniform identifiable hypergraph of order n and size m with maximum degree at most 3, then ${20\tau_D(H) \leq 12n + 3m}$ . Using this result, we then show that if G is a twin-free cubic graph on n vertices, then ${\gamma^{\rm IOC}(G) \leq 3n/4}$ . This bound is achieved, for example, by the hypercube.