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

A graph $G$ is called $C_4$-free if it does not contain the cycle $C_4$ as aninduced subgraph. Hubenko, Solymosi and the first author proved (answering aquestion of Erd\H os) a peculiar property of $C_4$-free graphs: $C_4$ graphswith $n$ vertices and average degree at least $cn$ contain a complete subgraph(clique) of size at least $c'n$ (with $c'= 0.1c^2n$). We prove here betterbounds (${c^2n\over 2+c}$ in general and $(c-1/3)n$ when $ c \le 0.733$) fromthe stronger assumption that the $C_4$-free graphs have minimum degree at least$cn$. Our main result is a theorem for regular graphs, conjectured in the papermentioned above: $2k$-regular $C_4$-free graphs on $4k+1$ vertices contain aclique of size $k+1$. This is best possible shown by the $k$-th power of thecycle $C_{4k+1}$.

Cliques in C_4-free graphs of large minimum degree