On a refinement of the Ahlswede--Katona Theorem

TL;DR

This work refines the Ahlswede--Katona result for the 2-edge star (cherry) by imposing a large independent set with high minimum degree and analyzing the asymptotic maximum of $N(S_2,G)$ via the Zagreb index $Z_{1}(G)$. The authors introduce refined extremal constructions and a bipartite counting framework, deriving tight upper bounds on the $S_2$-density $ ho(S_2,G)$ in specific regimes of the edge density $ ho$ and independence parameter $ ho,eta$, with precise formulas such as $I(S_2, ho, frac{ ext{alpha}}{n}, frac{ ext{alpha}}{n})= ext{alpha}^3+( ho- ext{alpha}^2)\sqrt{ ho+ ext{alpha}^2}$ for certain ranges of $ ho$ and $ ext{alpha}$. The results connect to hypergraph Turán problems in the $ imes ext{l}_2$-norm and are established through a combination of Zagreb-index analysis, extremal bipartite graph bounds, and structured case analyses that pinpoint near-extremal graphs. The findings provide exact asymptotics and structural insight for refined cherry-counting problems, with constructions $S(n,m)$, $C(n,m)$, $G_{1}$, $G_{2}$, $B_1$, and $B_2$ underpinning the lower and upper bounds.

Abstract

A classical theorem of Ahlswede and Katona determines the maximum density of the $2$-edge star in a graph with a given edge density. Motivated by its application in hypergraph Turán problems, we establish a refinement of their result under the additional assumption that the graph contains a large independent set in which every vertex has high degree.

Theorems & Definitions (29)

• Theorem 1.1: Ahlswede--Katona AK78
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Ahlswede--Katona AK78
• Theorem 1.7
• Theorem 1.8
• proof : Proof of Theorem \ref{['Thm:biparite-ell-k-Z_1(G)']}
• Claim 3.1
• proof : Proof of Claim \ref{['Claim:complete-bipartite-graph-bi']}
• Claim 3.2