7 problems

Fischer's conjecture. If $\delta'(G)\geq(1-1/k)n$, then $G$ has a $K_k$-factor. This is a partite analogue of Hajnal--Szemerédi-type clique-factor results; the conjecture concerns…

Let $r$ and $s$ be integers with $1<s<r$, and let $n$ be divisible by $r$. Define … For a graph $G$, let $G^p=G\cup G_{n,p}$ denote its randomly perturbed version, and let a graph…

Let $r\geqslant 3$ be constant, let $p\leqslant n^{-2/r+o(1)}$, and let $m=\big\lfloor\binom{n}{r}p^{\binom{r}{2}}\big\rfloor$. Write $F_r(G(n,p))$ for the number of $K_r$-factors…

For integers $k,r\geq 1$, let $f(k,r)$ be the smallest integer such that every $k$-connected graph $G$ whose order is divisible by $r$ has a $K_r$-factor in $G^{f(k,r)}$. The 3-con…

Let $r\geq 2$ be an integer, and let $G$ be a 2-connected graph whose order is divisible by $r$. A $K_r$-factor is a spanning subgraph whose components are copies of the complete g…

Modified Fischer conjecture. Fischer's conjecture should hold for balanced $r$-partite graphs, except in the examples described by Catlin and by Keevash and Mycroft: when $rn/k$ is…

Let $t$ and $n$ be positive integers, and let $\textbf{G}=\{G_1,G_2,\ldots,G_{\frac{n}{t}\binom{t}{2}}\}$ be an $n$-vertex graph system. Rainbow Hajnal--Szemerédi conjecture. If ……