10 problems

Let $\mathcal{G}_5$ be the class of planar graphs of girth at least $5$, and let $n$ denote the order of such a graph. An induced forest is an induced subgraph that is a forest. Ko…

Akiyama–Watanabe conjecture. If $G$ is bipartite, then

Let $G$ be a bipartite planar graph, and let an induced forest be a vertex-induced subgraph of $G$ that is a forest. Akiyama–Watanabe and Albertson–Rhaas conjecture. Every bipartit…

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

Let $\mathcal{B}$ be a balanced bipartite graph of order $2n$, with bipartition into two parts $V_1$ and $V_2$ each containing $n$ vertices. The minimum degree of $\mathcal{B}$ is…

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 planar graph, and let an induced linear forest be a vertex-induced forest whose maximum degree is at most $2$. Chappel's conjecture. Every planar graph admits an induc…

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 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…