AI 中文总结
本文证明了局部团覆盖数与色数之和不超过顶点数加一的猜想，并给出了达到等号时图的结构性质及诱导匹配的改进界。
AI 中文摘要
局部团覆盖数$lcc(G)$是$G$的边团覆盖的最小度数。我们证明了对于每个有限简单图，猜想的不等式$lcc(G)+\chi(G)\le |V(G)|+1$成立。证明给出了端点覆盖估计的独立集改进，并将所得构造应用于诱导四顶点路径。我们进一步证明，该覆盖可以被选择使得每个非全向顶点的度数至多为$|V(G)|-\chi(G)$。因此，每个达到等号的图都有一个全向顶点。更强的结论通过将最小阶反例归约为素数双临界图并细化诱导路径扩展来获得。我们还证明了大小为$m\ge2$的诱导匹配产生$lcc(G)+\chi(G)\le |V(G)|+3-m$，当$m\ge3$时改进了一般界，并确定了等号对删除诱导$2K_2$所施加的限制。
英文摘要
The local clique cover number $lcc(G)$ is the minimum valency of an edge-clique cover of $G$. We prove the conjectured inequality $lcc(G)+χ(G)\le |V(G)|+1$ for every finite simple graph. The proof gives an independent set improvement of an endpoint-cover estimate and applies the resulting construction to an induced four-vertex path. We further prove that the cover can be chosen so that every nonuniversal vertex has valency at most $|V(G)|-χ(G)$. Consequently, every graph attaining equality has a universal vertex. The stronger statement follows by reducing a counterexample of minimum order to a prime double-critical graph and refining the induced path extension. We also show that an induced matching of size $m\ge2$ yields $lcc(G)+χ(G)\le |V(G)|+3-m$, which improves the general bound when $m\ge3$, and determine the restrictions imposed by equality on deletion of an induced $2K_2$.
