Exact extremizers for $\ell_p$-judicious bisections of triangle-free graphs

Abstract

Balogh, Clemen and Lidický asked how large the minimum $\ell_p$-norm of the internal-edge counts over balanced partitions of an $n$-vertex triangle-free graph can be. For each fixed $2<p<\infty$ and all sufficiently large even $n$, we determine this value exactly for bisections and characterize all extremal graphs. They are precisely the complete bipartite graphs $K_{a,n-a}$ for which $a\le n/2$ maximizes an explicit integer objective. All maximizing values of $a$ lie in an interval of length bounded in terms of $p$ alone. We also prove quantitative stability for the same range of $p$ and $n$: for sufficiently small $δ\ge 0$, every $n$-vertex triangle-free graph with bisection value at least the asymptotic optimum minus $δn^2$ is within $O_p(δn^2+n)$ edge edits of a complete bipartite graph with the optimal limiting class ratio.