[论文解读] The exact value of $c_1(K_{2,n})$

Hiroaki Mori|arXiv (Cornell University)|Feb 27, 2026

Graph Labeling and Dimension Problems被引用 0

一句话总结

论文确定将完整二分图 K_{2,n} 的最短路径距离嵌入到 ℓ1 的精确失真度，给出 c1(K_{2,n})=\frac{3\lceil n/2\rceil-2}{2\lceil n/2\rceil-1}，适用于所有正整数 n。

ABSTRACT

For a graph $G$, let $c_1(G)$ be the largest distortion necessary to embed any shortest-path metric on $G$ into $\ell_1$, and for any natural number $n,m\in\mathbb{N}$, denote $K_{n,m}$ as the complete bipartite graph. In this note, we caculate the value of $c_1(K_{2,n})$, more precisely we prove $c_1(K_{2,n})=\frac{3k-2}{2k-1}$ where $k=\lceil\frac{n}{2} ceil$.

研究动机与目标

• Motivate the study of embedding shortest-path metrics into $\\ell_1$ for bipartite graphs and quantify the flow-cut gap via $c_1(G)$.
• Determine the exact value of $c_1(K_{2,n})$ for all positive integers $n$.
• Extend previous results showing asymptotic behavior $\\lim_{n\\to\\infty} c_1(K_{2,n})=3/2$ and identify specific small-n values.
• Provide tight lower and upper bounds by leveraging hypermetric inequalities and structured graph constructions.

提出的方法

• Use hypermetric inequality to derive a lower bound: for $K_{2,2k+1}$ embed with distortion $D$ implies $D\\ge\\frac{3k+1}{2k+1}$.
• Construct bipartite graphs $K_{2,2k}^{\\ell}$ by subdividing edges to obtain $2k$ internally disjoint $2\\ell$-length paths between the two side-vertices.
• Combine two random-cut schemes to produce an $\\ell_1$ embedding with distortion $\\frac{3k-2}{2k-1}$ for $K_{2,2k}^{\\ell}$, and verify case-by-case that the distortion bound is achieved across pairs of vertices.
• Apply a compactness argument to reduce the upper-bound analysis to the subdivided graphs, enabling explicit embedding via cut metrics.
• Show that the resulting embedding distortion matches the lower bound, hence $c_1(K_{2,2k})\\le\\frac{3k-2}{2k-1}$.
• Conclude by translating results back to $K_{2,n}$ with $k=\\lceil n/2\\rceil$ and all $n$.

实验结果

研究问题

• RQ1 What is the exact value of the distortion constant $c_1(K_{2,n})$ for every positive integer $n$?
• RQ2 How do hypermetric inequalities constrain embeddings of $K_{2,n}$ into $\\ell_1$?
• RQ3 Can the distortion upper bound be achieved via explicit random-cut constructions for subdivided bipartite graphs?
• RQ4 What is the asymptotic behavior of $c_1(K_{2,n})$ as $n$ grows large?
• RQ5 How do small-n values (e.g., $n=1,2,3$) fit into the general formula?

主要发现

• The exact value is $c_1(K_{2,n})=\\frac{3\\lceil n/2\\rceil-2}{2\\lceil n/2\\rceil-1}$ for all positive $n$, proven via matching lower and upper bounds.
• A lower bound is established for odd $n=2k+1$ as $c_1(K_{2,2k+1})\\ge\\frac{3k+1}{2k+1}$.
• An explicit embedding with distortion $\\frac{3\\lceil n/2\\rceil-2}{2\\lceil n/2\\rceil-1}$ is constructed for even $n$, using subdivided graphs $K_{2,2k}^{\\ell}$ and two random-cut schemes.
• Thus, the paper provides the first exact $c_1(G)$ value for an infinite family of graphs (excluding the trivial $c_1(G)=1$ cases).
• The result strengthens prior work showing $\\lim_{n\\to\\infty} c_1(K_{2,n})=3/2$ and clarifies finite-n behavior.
• The exact value matches both the lower and upper bound analyses across all $n$.