The edge clique cover number $\text{ecc}\left(G\right)$ of a graph $G$ is the size of the smallest collection of complete subgraphs whose union covers all edges of $G$. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if $G$ is claw‐free, then $\text{ecc}\left(G\right)$ is bounded above by its order (denoted $n$). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in $n$ for $\text{ecc}\left(G\right)$ for such graphs, improving upon the bou nd of $O(n^{4∕3}{\log }^{1∕3} n)$ due to Javadi, Maleki, and Omoomi. More precisely we prove that $\text{ecc}\left(G\right)$ is at most the minimum of $n+\delta \left(G\right)$ and $2n-\mathrm{\Omega }\left(\sqrt{n\log n}\right)$, where $\delta \left(G\right)$ is the minimum degree of $G$. In the fractional version of the problem, we improve these upper bounds to $\frac{3}{2}n$. We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for $\text{ecc}\left(G\right)$) equals $n$, and when $G$ contains no induced subgraph isomorphic to $H$ where $H$ is any fixed graph of order 4.

中文翻译：

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独立性为第二的图中的边缘集团

边缘集团封面编号 $\text{心电图}（G）$ 图的 $G$ 是完整的子图的最小集合的大小，该子图的并集覆盖了 $G$。Chen，Jacobson，Kézdy，Lehel，Scheinerman和Wang在2000年推测，如果$G$ 没有爪子，那么 $\text{心电图}（G）$ 在上方受其顺序限制（表示为 $ñ$）。最近，Javadi和Hajebi验证了独立数至少为3的无爪图的猜想。我们研究具有第二独立性的图的边缘团覆盖数，这些图必须是无爪的。我们给出第一个已知的线性界的证明$ñ$ 为了 $\text{心电图}（G）$ 对于此类图形，改进了 $Ø（{ñ}^{4∕3}{\mathrm{日志}}^{\mathrm{1个}∕3} ñ）$由于Javadi，Maleki和Omoomi。更确切地说，我们证明$\text{心电图}（G）$ 最多是 $ñ+\delta （G）$ 和 $\mathrm{2个}ñ-\mathrm{\Omega }（\sqrt{ñ\mathrm{日志} ñ}）$， 在哪里 $\delta （G）$ 是最小程度 $G$。在问题的分数形式中，我们将这些上限提高为$\frac{3}{\mathrm{2个}}ñ$。我们还验证了某些特定子族的猜想，例如，当相对于集团的边装箱数（$\text{心电图}（G）$） 等于 $ñ$， 什么时候 $G$ 不包含与之同构的诱导子图 $H$ 在哪里 $H$ 是4阶的任何固定图。