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A NOTE ON JUDICIOUS BISECTIONS OF GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
- Published online by Cambridge University Press:
- 08 March 2024, pp. 401-408
- Print publication:
- December 2024
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- Article
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Let G be a graph with m edges, minimum degree
$\delta $ and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan et al. [‘Bisections of graphs without short cycles’, Combinatorics, Probability and Computing 27(1) (2018), 44–59] showed that if (i) G is 
$2$-connected, or (ii) 
$\delta \ge 3$, or (iii) 
$\delta \ge 2$ and the girth of G is at least 5, then G admits a bisection such that 
$\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$, where 
$e(V_i)$ denotes the number of edges of G with both ends in 
$V_i$. Let 
$s\ge 2$ be an integer. In this note, we prove that if 
$\delta \ge 2s-1$ and G contains no 
$K_{2,s}$ as a subgraph, then G admits a bisection such that 
$\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$.
