Generalized transversals, generalized vertex covers and node-fault-tolerance in graphs

Abstract

The transversal number τ(G) of a hypergraph G is the minimum cardinality of a set of vertices that intersects all edges of G. For r≥1 define fr=supτ(G)|V(G)|+|E(G)|, where G ranges over all r-uniform hypergraphs. In 1990 Alon proved that fr=(1+o(1))lnrr. We consider the following generalization of this problem. Given an r-uniform hypergraph H, the H-transversal number τH(G) of a hypergraph G is the minimum cardinality of a set of vertices that intersects the vertex set of every copy of H in G. Define fr(H)=supτH(G)|V(G)|+|E(G)|, where G ranges over all r-uniform hypergraphs. We prove that f2(Kq)=12q−1, f2(P3)=1∕4, f2(P4)=1∕5, f2(C4)=1∕6, and 152q+1≤f2(Cq)≤f2(Pq)≤1∕5 for q≥5. In order to prove these results we establish a connection between H-transversals and the node-fault-tolerance in graphs. Furthermore, we derive some bounds on the H-independence number.