7 problems

Let $H$ be a graph with chromatic number $\ell:=\chi(H)$ and bandwidth measuring that $H$ has not too strong expansion properties. Let $G$ be a graph with minimum degree…

Let $P_n^k$ be the $k$-th power of the path on $n$ vertices, and let $\deg_G(v)$ denote the degree of $v$ in $G$. Hamiltonicity conjecture for path powers. Let $n\geq4$ and…

Let $G_{m,n}$ be the rectangular grid graph, and let $k$ be a positive integer with $k<n/2$. Reducing the bandwidth of $G_{m,n}$ by $k$ means deleting edges so that the resulting g…

Maturana–Rashmi's conjecture. Under the Uniform Cost Assumption, every stable linear MDS $(n^{I},k^{I}={\lambda}^{F}k^{F};n^{F},k^{F})$ convertible code with $r^{F}<r^{I}$ and…

Cluster-size conjecture. Theorem remains valid for $H$ with clusters of size at most

Let $C_n^k$ be the $k$-th power of the cycle on $n$ vertices, and let $\delta(G)$ denote the minimum degree of a graph $G$. ELW's Hamiltonicity conjecture. For all integers…

For each $\gamma>0$, $\Delta\ge2$, and $k\ge1$, there should exist constants $\beta^\ast>0$ and $C^\ast>0$ such that, asymptotically almost surely for $\Gamma=G(n,p)$ whenever…