Large Cuts with Local Algorithms on Triangle-Free Graphs

Juho Hirvonen*, Joel Rybicki, Stefan Schmid, Jukka Suomela

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

12 Citations (Scopus)
54 Downloads (Pure)


Let G be a d-regular triangle-free graph with in edges. We present an algorithm which finds a cut in G with at least (1/2 + 0.28125/root d)rn edges in expectation, improving upon Shearer's classic result. In particular, this implies that any d-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of G.

Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed d, there exists a weighted neighbourhood graph N-d such that there is a one-to-one correspondence between heavy cuts of N-d and randomised local algorithms that find large cuts in any d-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in d-regular graphs: we can compute the optimal cut of N-d to attain a lower bound on the maximum cut size of any d-regular triangle-free graph.

Original languageEnglish
Article numberP4.21
Pages (from-to)1-20
Number of pages20
JournalThe Electronic Journal of Combinatorics
Issue number4
Publication statusPublished - 20 Oct 2017
MoE publication typeA1 Journal article-refereed


  • graph theory
  • regular graphs
  • cuts


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