Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Ning, Bo
Institution: Nankai University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2884061123, G505044266, G7484038684
OpenAlex ID: W7213599420