Some Results on Weighted Graphs without Induced Cycles of Nonpositive Weights

Let G be a simple graph; assume that a mapping assigns integers as weights to the edges of G such that for each induced subgraph that is a cycle, the sum of all weights assigned to its edges is positive; let σ be the sum of weights of all edges of G. It has been proved (Vijayakumar, Discrete Math 311(14):1385–1387, 2011) that (1) if G is 2-connected and the weight of each edge is not more than 1, then σ is positive. It has been conjectured (Xu, Discrete Math 309(4):1007–1012, 2009) that (2) if the minimum degree of G is 3 and the weight of each edge is ±1, then σ > 0. In this article, we prove a generalization of (1) and using this, we settle a vast generalization of (2).