AI 中文总结
本文研究一类广义弦图（有界单纯亏量图），证明其最大团划分数公式并确定极图，回答Erdős等的问题，同时给出结构稳定性与有限阶定理。
AI 中文摘要
图的团划分数是将边集划分成若干完全子图的边集所需的最少完全子图数。我们研究这样的图：在每个导出子图中，且在任意给定的团之外，存在某个顶点，其邻域在删除至多$s$个顶点后成为团。$s=0$的情形恰为弦图类。对每个固定的$s$，我们证明：对所有足够大的阶数$n$，最大团划分数为$\lfloor(n+s)(n+s+1)/6\rfloor-\binom{s+1}{2}$，并确定所有达到等号的图。同一表达式在任意阶数下都是上界，至多相差一个仅依赖于$s$的加性常数。特别地，每个弦图的团划分数至多为$n^2/6+n/6+O(1)$，回答了Erdős、Ordman和Zalcstein的一个问题。我们还证明了次线性亏量的结构稳定性，以及整数符号团泛函的一个尖锐有限阶定理。证明结合了带符号分数局部化与边不相交三角形构造；仅需定性的分数打包逼近。
英文摘要
The clique partition number of a graph is the minimum number of complete subgraphs whose edge sets partition its edge set. We study graphs in which, in every induced subgraph and outside every prescribed clique, some vertex has a neighbourhood that becomes a clique after deleting at most $s$ vertices. The case $s=0$ is exactly the class of chordal graphs. For each fixed $s$, we prove that the maximum clique partition number at all sufficiently large orders $n$ is $\lfloor(n+s)(n+s+1)/6\rfloor-\binom{s+1}{2}$, and determine all equality graphs. The same expression is an upper bound up to an additive constant depending only on $s$ at every order. In particular, every chordal graph has clique partition number at most $n^2/6+n/6+O(1)$, answering a question of Erdős, Ordman and Zalcstein. We also prove structural stability for sublinear defect and a sharp finite-order theorem for integer signed clique functionals. The proof combines signed fractional localization with an edge-disjoint triangle construction; only a qualitative fractional-packing approximation is required.
Comments24 pages
