Proof of the Kalai-Meshulam conjecture

Maria Chudnovsky; Alex Scott; Paul Seymour; Sophie Spirkl

Proof of the Kalai-Meshulam conjecture

Combinatorics 2018-10-02 v1

Authors: Maria Chudnovsky , Alex Scott , Paul Seymour , Sophie Spirkl

Abstract

Let $G$ be a graph, and let $f_G$ be the sum of $(-1)^{|A|}$ , over all stable sets $A$ . If $G$ is a cycle with length divisible by three, then $f_G= \pm 2$ . Motivated by topological considerations, G. Kalai and R. Meshulam made the conjecture that,if no induced cycle of a graph $G$ has length divisible by three, then $|f_G|\le 1$ . We prove this conjecture.

Keywords

graph theory stable sets induced cycles Kalai-Meshulam conjecture combinatorics

Cite

@article{arxiv.1810.00065,
  title  = {Proof of the Kalai-Meshulam conjecture},
  author = {Maria Chudnovsky and Alex Scott and Paul Seymour and Sophie Spirkl},
  journal= {arXiv preprint arXiv:1810.00065},
  year   = {2018}
}

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