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arXiv · 2609.20305

On the difference between clique partition and clique covering numbers

Bo Ning
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Abstract

For a graph \(G\), let \(\operatorname{cp}(G)\) and \(\operatorname{cc}(G)\) be the minimum numbers of cliques in an edge partition and a clique cover of \(G\), respectively. Set $σ_n = \max_{\lvert V(G)\rvert=n} \bigl(\operatorname{cp}(G)-\operatorname{cc}(G)\bigr), d_n=\left\lfloor\frac{n^2}{4}\right\rfloor-σ_n.$ In 1983, Erdős, Faudree, and Ordman asked whether \(d_n=O(n)\). Caccetta, Erdős, Ordman, and Pullman previously constructed graphs showing \(d_n=O(n^{3/2})\). We prove that $d_n=Θ(n^{4/3}),$ thereby determining the correct order of the deficit and answering their question in the negative.

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Bo Ning. 2026-08-01. On the difference between clique partition and clique covering numbers. https://arxiv.org/abs/2609.20305

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