A property of weighted graphs without induced cycles of nonpositive weights

It 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.