Abstract
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).
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References
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Xu B.: On signed cycle domination in graphs. Discrete Math. 309(4), 1007–1012 (2009)
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Vijayakumar, G.R. Some Results on Weighted Graphs without Induced Cycles of Nonpositive Weights. Graphs and Combinatorics 29, 1101–1111 (2013). https://doi.org/10.1007/s00373-012-1176-9
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DOI: https://doi.org/10.1007/s00373-012-1176-9