9 problems

Maximum independent edge set conjecture.

Let $P_m$ and $P_n$ be path graphs, and consider their Cartesian product. The interior vertices are those corresponding to interior vertices of the paths, while the boundary consis…

Let $P_n$ be the path on $n$ vertices, and let $\overset{2}{\uplus}~P_n$ denote its double blow-up, with corresponding vertices $(0,u)$ and $(1,u)$ for each vertex $u$ of $P_n$. Le…

Sum-diameter conjecture for paths. For $n\geq 3$, we have

Spum conjecture for paths. For $n\geq 8$, we have

Let $t$ be a positive integer, let $P_{2t}$ denote the path of length $2t$, and let $\operatorname{ex}^{-1}(k,H)$ denote the inverse Turán number of $H$, namely the maximum number…

Propp's conjecture. Around each orbit of independent sets under the action of $\varphi$, the number of times vertex $i$ occurs is the same as the number of times vertex $N-i-1$ occ…

Let $P_n$ denote the path graph on $n$ vertices, and let $k$ be a positive integer. A graph is $k$-super graceful when it has a $k$-super graceful labeling, as defined in the paper…

Rodríguez-Velázquez et al.'s conjecture. The metric dimension of the strong product of the two paths is