9 problems

Lower-bound conjecture. If $G$ has minimum degree at least $2$, then

Let $q>7$ be a prime power, and let $\mathcal{P}(q^2)$ denote the Paley graph on the finite field of order $q^2$. Let $\tau(G)$ denote the maximum order of an induced forest in a g…

Let $G$ be a planar graph, and let $\fva(G)$ denote its fractional vertex-arboricity. Fractional Albertson–Berman conjecture. Every planar graph has fractional vertex-arboricity at…

Let $G$ be a simple bipartite planar graph. Write $|G|=|V(G)|$, and let $a(G)$ denote the maximum number of vertices in an induced forest of $G$. Akiyama–Watanabe conjecture. … The…

Kowalik et al.'s conjecture. The graph $G$ admits an induced forest of order at least

Akiyama–Watanabe–Albertson–Haas conjecture. Every bipartite planar graph of order $n$ admits an induced forest of order at least

Let $G$ be a graph with maximum degree $\Delta$. An equitable partition is a partition of the vertex set into parts whose sizes differ by at most one, and an induced forest is a ve…

Let $G$ be a bipartite planar graph of order $n$. An induced forest of $G$ is a vertex-induced subgraph that is acyclic. Akiyama–Watanabe and Albertson–Rhaas conjecture. Every bipa…

Let $G$ be a planar graph of order $n$. An induced forest of $G$ is a vertex-induced subgraph that is acyclic. Albertson and Berman's conjecture. Every planar graph of order $n$ ad…