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 vertex-coloring of a hypergraph is conflict-free, if each edge contains avertex whose color is not repeated on any other vertex of that edge. Let $f(r,\Delta)$ be the smallest integer $k$ such that each $r$-uniform hypergraph ofmaximum vertex degree $\Delta$ has a conflict-free coloring with at most $k$colors. As shown by Tardos and Pach, similarly to a classical Brooks' typetheorem for hypergraphs, $f(r, \Delta)\leq \Delta+1$. Compared to Brooks'theorem, according to which there is only a couple of graphs/hypergraphs thatattain the $\Delta+1$ bound, we show that there are several infinite classes ofuniform hypergraphs for which the upper bound is attained. We provide bounds on$f(r, \Delta)$ in terms of~$\Delta$ for large~$\Delta$ and establish theconnection between conflict-free colorings and so-called $\{t, r-t\}$-factorsin $r$-regular graphs. Here, a $\{t, r-t\}$-factor is a factor in which eachdegree is either $t$ or $r-t$. Among others, we disprove a conjecture of Akbariand Kano~[Graphs and Combinatorics 30(4):821--826, 2014] stating that there isa $\{t,r-t\}$-factor in every $r$-regular graph for odd $r$ and any odd$t<\frac{r}{3}$.

Brooks Type Results for Conflict-Free Colorings and {a, b}-factors in graphs