AI 中文总结
本文针对大图的团分解与覆盖问题，解决了Erdős等人提出的两个团相关猜想，确定了等号成立的情况，采用线性规划对偶性、图删除引理等方法完成证明。
AI 中文摘要
1966年，Erdős、Goodman和Pósa证明，每个n顶点图G都可使用至多⌊n²/4⌋个团覆盖其边，该界由平衡完全二部图达到。Erdős提出如下强化猜想：每个n顶点图G都可分解为若干团，总代价至多为⌊n²/4⌋，其中每个i-团的代价为i-1。Dau、Milenkovic和Puleo提出另一推广猜想：对所有t≥4，每个n顶点图G都可使用至多∏ⱼ₌₀ᵗ⁻¹⌊(n+j)/t⌋个团覆盖其t-团。Balogh、He、Krueger、Nguyen和Wigal证明了这些猜想的渐近版本和分数版本。本文针对大n求解上述两个猜想，并确定等号成立的情况。对Erdős猜想的证明结合了线性规划对偶性、生成树多面体、Mantel定理的加权稳定形式，以及围绕几乎平衡二划分的显式分解算法；对t-团覆盖猜想的证明则结合了图删除引理、广义Turán稳定定理，以及对接近Tₙ,ₜ的图的精确覆盖构造。
英文摘要
In 1966, Erdős, Goodman, and Pósa showed that every $n$-vertex graph $G$ admits a cover of its edges using at most $\lfloor \frac{n^2}{4}\rfloor$ cliques, with tightness witnessed by the balanced complete bipartite graph. Erdős suggested the following strengthening: every $n$-vertex graph $G$ admits an edge decomposition into cliques with total cost at most $\lfloor \frac{n^2}{4}\rfloor$, where each $i$-clique has cost $i-1$. There is another generalization conjectured by Dau, Milenkovic and Puleo: for every $t\ge4$, every $n$-vertex graph $G$ admits a cover of its $t$-cliques using at most $\prod_{j=0}^{t-1}\left\lfloor\frac{n+j}{t}\right\rfloor$ cliques. Balogh, He, Krueger, Nguyen and Wigal proved asymptotic and fractional versions of these conjectures. We solve both conjectures for large $n$ and identify the equality cases. Our proof of Erdős's conjecture combines linear programming duality and the spanning-forest polytope with a weighted stability form of Mantel's theorem, followed by explicit decomposition algorithms around an almost balanced bipartition. For the $t$-clique-cover conjecture, we combine graph removal lemma and generalized Turán stability with an exact covering construction for graphs close to $T_{n,t}$.
Comments35 pages
