A property of weighted graphs without induced cycles of nonpositive weights

AbstractIt has been conjectured [B. Xu, On signed cycle domination in graphs, Discrete Math. 309 (4) (2009) 1007–1012] that if there is a mapping from the edge set of a 2-connected graph G to {−1,1} such that for each induced subgraph, that is a cycle, the sum of all numbers assigned to its edges by this mapping is positive, then the number of all those edges of G to which 1 is assigned, is more than the number of all other edges of G. This conjecture follows from the main result of this note: If a mapping assigns integers as weights to the edges of a 2-connected graph G such that for each edge, its weight is not more than 1 and for each cycle which is an induced subgraph of G, the sum of all weights of its edges is positive, then the sum of all weights of the edges of G also is positive. A simple corollary of this result is the following: If ϕ is a mapping from the edge set of a 2-connected graph G to a set of real numbers such that for each cycle C of G, ∑e∈E(C)ϕ(e)>0, then ∑e∈E(G)ϕ(e) also is positive