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

Let $G$ be a graph without 4-cycles and 5-cycles. We show that the problem todetermine whether $G$ is $(0,k)$-colorable is NP-complete for each positiveinteger $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without4-cycles and 5-cycles for each positive integer $k.$ Finally, we prove that $G$is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4), (3,5),$ and $(2,9).$

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles