Universally signable graphs

Abstract

In a graph, a chordless cycle of length greater than three is called a hole. Let γ be a {0, 1} vector whose entries are in one-to-one correspondence with the holes of a graphG. We characterize graphs for which, for all choices of the vector γ, we can pick a subsetF of the edge set ofG such that |F ∪H| эγH (mod 2), for all holesH ofG and |F ∪T| э 1 for all trianglesT ofG. We call these graphsuniversally signable. The subsetF of edges is said to be labelledodd. All other edges are said to be labelledeven. Clearly graphs with no holes (triangulated graphs) are universally signable with a labelling of odd on all edges, for all choices of the vector γ. We give a decomposition theorem which leads to a good characterization of graphs that are universally signable. This is a generalization of a theorem due to Hajnal and Surányi [3] for triangulated graphs.