Abstract
Let \(G=(V,E,w)\) be a weighted graph. The independence number of a weighted graph G is the maximum w(S) taken over all independent sets S of G. By a well-known method, we can show that the independence number of G is at least \(\sum _{v\in V} \frac{w(v)}{1+d(v)}\), where d(v) is the degree of v. In this paper, we consider the independence numbers of weighted graphs with forbidden cycles. For a graph G, the odd girth of G is the smallest length of an odd cycle in G. We show that if G is a weighted graph with odd girth \(2m+1\) for \(m\ge 3\), then the independence number of G is at least \(c\left( \sum _{v}w(v)d(v)^{1/(m-2)}\right) ^{(m-2)/(m-1)}\), where c is a constant.
Supported in part by NSFC (12101156).
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Wang, Y., Li, Y. (2021). The Independence Numbers of Weighted Graphs with Forbidden Cycles. In: Wu, W., Du, H. (eds) Algorithmic Aspects in Information and Management. AAIM 2021. Lecture Notes in Computer Science(), vol 13153. Springer, Cham. https://doi.org/10.1007/978-3-030-93176-6_33
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DOI: https://doi.org/10.1007/978-3-030-93176-6_33
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