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Let us have a connected graph $\mathcal{G}(V,E)$ where $V$ are the vertices and $E$ the edges. Define the maximum-cut as $M(\mathcal{G})$, which is a division of the vertices into two sets with a maximal number of edges between the sets.

In general we have $E/2 \leq M(\mathcal{G}) \leq E$ and I am interested in the case where there is no 'good' max cut as the graph becomes increasingly large, i.e. $M(\mathcal{G})/E = 1/2$ as we take $|V| \rightarrow \infty$.

This is the case for a number of `dense' graphs (i.e. those with a number of edges $|E|$ quadratic in the number of vertices $|V|$). Some examples would be the complete graph or an Erdos-Renyi with a constant $p$.

My question is: do there exist `sparse' graphs (i.e. those with a number of edges linear in the number of vertices) where $M(\mathcal{G})/E = 1/2$ as $|V| \rightarrow \infty$ or can it be proven that $M(\mathcal{G})/E > 1/2$ for such graphs? What is the sparse graph with the smallest $M(\mathcal{G})$?

EducationalFerret
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  • How about a disjoint union of a bunch of (large) constant size complete graphs? The max cut would have to split each of these in half (and get about half the edges). – dbal Sep 29 '21 at 16:30
  • Nice example! What if I want the graph to be connected? Would adding a single edge between each of those complete graphs ruin it or not? – EducationalFerret Sep 29 '21 at 16:59
  • It wouldn't really "ruin it". Already this construction has max cut (1/2)(k/(k-1)) which is "off" by about 1/k (where k is the size of the complete graphs). Adding n/k extra edges having those all included wouldn't change that really – dbal Sep 29 '21 at 18:11
  • Hmm, but in order for the graph to be sparse we would need k to be a fixed parameter and n (the number of complete graphs) to be the a variable which we can take to infinity. In that case the number of added edges would be be significant and, I think, would make $M(\mathcal{G})/E > 1/2$ as $n \rightarrow \infty$. – EducationalFerret Sep 29 '21 at 18:34
  • Right, the first construction only gives M(G) < 1/2 + 1/k (approximately). So yes, it is off. But you can get a statement like, for any epsilon>0, there is a sequence of sparse graphs (where the sparseness gets worse as epsilon -> 0) which satisfy M(G)/E < .5 + epsilon. Maybe someone has a better idea – dbal Sep 29 '21 at 18:39

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