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

Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285]conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$and minimum degree $\delta\geq 2$ is at most$\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta} + O(1)$ for every $r\ge2$, if $\delta$ is a multiple of $(r-1)(3r+2)$. For every $r>1$ and $\delta\ge2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $\delta$ anddiameter $\frac{(6r-5)n}{(2r-1)\delta+2r-3}+O(1)$, which are counterexamples tothe conjecture for every $r>1$ and $\delta>2(r-1)(3r+2)(2r-3)$. The rest of thepaper proves positive results under a stronger hypothesis, $k$-colorability,instead of being $K_{k+1}$-free. We show that the diameter of connected$k$-colorable graphs with minimum degree $\geq \delta$ and order $n$ is at most$\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}+O(1)$, while for $k=3$, it is atmost $\frac{57n}{23\delta}+O\left(1\right)$.

On the maximum diameter of $k$-colorable graphs