The Half-integral Erdös-Pósa Property for Non-null Cycles

A Group Labeled Graph is a pair (G,Λ) where G is an oriented graph and Λ is a mapping from the arcs of G to elements of a group. A (not necessarily directed) cycle C is called non-null if for any cyclic ordering of the arcs in C, the group element obtained by ‘adding’ the labels on forward arcs and ‘subtracting’ the labels on reverse arcs is not the identity element of the group. Non-null cycles in group labeled graphs generalize several well-known graph structures, including odd cycles. A family F of graphs has the Erdős-Pósa property, if for every integer k there exists an integer f(k,F) such that every graph G contains either k vertex-disjoint subgraphs each isomorphic to a graph in F or a set at most f(k,F) vertices intersecting every subgraph isomorphic to a graph in F . In this paper, we prove that non-null cycles on Group Labeled Graphs have the 12 -integral Erdős-Pósa property. That is, there is a function f : N → N such that for any k ∈ N, any group labeled graph (G,Λ) has a set of k non-null cycles such that each vertex of G appears in at most two of these cycles or there is a set of at most f(k) vertices that intersects every non-null cycle. Since it is known that non-null cycles do not have the Erdős-Pósa property in general, a 1 2 -integral Erdős-Pósa result is the best one could hope for. ∗Department of Informatics, University of Bergen [email protected] †Algorithms and Complexity Group, TU Wien [email protected] ‡The Institute of Mathematical [email protected] ar X iv :1 70 3. 02 86 6v 1 [ cs .D M ] 8 M ar 2 01 7