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

Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an$F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that togethercover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$such that every $k$-graph $H$ on $n$ vertices with minimum codegree at least$t$ contains a perfect $F$-tiling. Mycroft (JCTA, 2016) determined theasymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$. Mycroftalso conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replacedby a constant that depends only on $F$. In this paper, we improve the errorterm of Mycroft's result to a sub-linear term when $F=K^3(m)$, the complete$3$-partite $3$-graph with each part of size $m$. We also show that thesub-linear term is tight for $K^3(2)$, {the result also provides another familyof counterexamples of Mycroft's conjecture (Gao, Han, Zhao (arXiv, 2016) gave afamily of counterexamples when $H$ is a $k$-partite $k$-graph with somerestrictions.)} Finally, we show that Mycroft's conjecture holds forgeneralized 4-cycle $C_4^3$ (the 3-graph on six vertices and four distinctedges $A, B, C, D$ with $A\cup B= C\cup D$ and $A\cap B=C\cap D=\emptyset$),i.e. we determine the exact value of $t_2(n, C_4^3)$.

Codegree conditions for tilling balanced complete $3$-partite $3$-graphs and generalized 4-cycles