7 problems

For every finite graph $G$ with order $n=|V(G)|$, size $m=|E(G)|$, and maximum degree $\Delta=\Delta(G)$, does the identifying vertex cover number of $G$ satisfy the upper-bound in…

Chordal graph k-path vertex cover conjecture.

Planar 3-path vertex cover conjecture.

Let $G$ be a very well covered graph, and for $s\in\mathbb{N}$ let $\mathcal{J}_s(G)$ denote its $s$-th jet graph. A graph is very well covered if all its minimal vertex covers hav…

Let $n,s,k$ be positive integers with $k\geq 2$ and $n\geq ks+k-1$. Let $H$ be a $k$-graph on vertex set $[n]$, and let $\nu(H)$ and $\tau(H)$ denote its matching number and vertex…

Let $G=(V,E)$ be a connected graph, let $H$ be a subgraph of $G$, and let $G[V\setminus V(H)]$ be the subgraph induced by the vertices outside $H$. Write $\delta(\cdot)$ for minimu…

Let $P(n,k)$ be the generalized Petersen graph, and let $\beta(P(n,k))$ denote the size of a minimum vertex cover. Behsaz–Hatami–Mahmoodian's conjecture. For all $n$ and $k$, … Sin…