奇环跨度缺陷：一个多项式下界和一个平方根上界
Odd-Cycle Span Defect: A Polynomial Lower Bound and a Square-Root Upper Bound
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中文总结 AI 辅助
本研究针对Erdos和Hajnal公开问题相关的图论函数F(N)，通过二元覆盖码构造等方法证明其多项式下界N^{1/6-o(1)}，结合多定理与最大独立集剥离得到平方根上界√(6N)，提升了该有限阶间隙的尺度下界。
中文摘要 AI 辅助
对于图$G$，令$ψ(G)=\text{max}\{χ(G[V(C)]):C$是$G$的一个奇环$\}$，当$G$是二部图时$ψ(G)=0$。对于正整数$N$，定义$F(N)=\text{max}\{χ(G)-ψ(G):|V(G)|\le N\}$。函数$F$衡量由Erdos和Hajnal的一个公开问题引出的有限阶加性间隙。我们证明$N^{1/6-o(1)}\le F(N)<\sqrt{6N}$。下界将Cameron-Clow路径着色构造提供的有限阶尺度从$\log N/\log\log N$提升到$N$的固定幂次。其证明从二元覆盖码$\mathcal{C}\subseteq\{0,1\}^p$构造调色板码图，并确立恒等式$χ(G)=2p+\ell-ρ(\mathcal{C})$和$ψ(G)=2p$。近中部汉明覆盖给出指数$1/6$。上界结合了Polavarapu连通性定理、Chvatal-Erdos哈密顿性定理和最大独立集剥离法。
英文摘要
For a graph $G$, let $ψ(G)=\max\{χ(G[V(C)]):C$ is an odd cycle of $G\}$, with $ψ(G)=0$ when $G$ is bipartite. For positive integers $N$, set $F(N)=\max\{χ(G)-ψ(G):|V(G)|\le N\}$. The function $F$ measures the finite-order additive gap arising from an open problem of Erdos and Hajnal. We prove $N^{1/6-o(1)}\le F(N)<\sqrt{6N}$. The lower bound raises the finite-order scale supplied by the Cameron-Clow path-colour construction from $\log N/\log\log N$ to a fixed power of $N$. Its proof constructs a palette-code graph from a binary covering code $\mathcal{C}\subseteq\{0,1\}^p$ and establishes the exact identities $χ(G)=2p+\ell-ρ(\mathcal{C})$ and $ψ(G)=2p$. Near-middle Hamming coverings yield the exponent $1/6$. The upper bound combines Polavarapu's connectivity theorem, the Chvatal-Erdos Hamiltonicity theorem, and maximum-independent-set stripping.
