English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$denote the order of the largest clique subdivision in $G$. Let H(n) be themaximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famousconjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for everygraph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. Thisconjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz furthershowed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for someabsolute constant $c>0$. In 1981 they conjectured that this bound is tight upto a constant factor in that there is some absolute constant $C$ such that$\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In thispaper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in ourproof, which might be of independent interest, is an estimate on the order ofthe largest clique subdivision which one can find in every graph on $n$vertices with independence number $\alpha$.

Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz