Abstract
Let G be a graph. A bisection of G is a bipartition of V(G) with \(V(G)=V_1\cup V_2\), \(V_1\cap V_2=\emptyset \) and \(||V_1|-|V_2||\le 1\). Bollobás and Scott conjectured that every graph admits a bisection such that for every vertex, its external degree is greater than or equal to its internal degree minus one. In this paper, we confirm this conjecture for some regular graphs. Our results extend a result given by Ban and Linial (J Graph Theory 83:5–18, 2016). We also give an upper bound of the maximum bisection of graphs.
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Funding
This work is supported by the National Natural Science Foundation of China (Nos. 12371349 and 12271169) and Zhejiang Provincial Natural Science Foundation of China (No. LY21A010002).
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This work is supported by the National Natural Science Foundation of China (Nos. 12371349 and 12271169) and Zhejiang Provincial Natural Science Foundation of China (No. LY21A010002).
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Yan, J., Chen, YH. Weak External Bisections of Regular Graphs. Graphs and Combinatorics 40, 63 (2024). https://doi.org/10.1007/s00373-024-02796-3
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DOI: https://doi.org/10.1007/s00373-024-02796-3