20 problems

Let $G$ be an optimal 1-planar graph, and let $h_{\min}(G)$ denote the minimum length of a hole in $G$. Minimum hole-length conjecture. … A construction gives infinitely many optim…

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. Ringel's conjecture. Every 1-planar graph is 6-colourable, that is, for every 1-plana…

Yang et al.'s conjecture. Every 1-planar graph is 6-choosable.

Extremal edge-count conjecture. For any odd $n\geq 5$,

Czap, Przybyło and Škrabul'áková's conjecture. The size of $G$ satisfies

For each integer $k\geq 1$, let $XH^k$ and $YH^k$ denote the graph families constructed in the paper. Maximality conjecture for $XH^k$ and $YH^k$. The graphs $XH^k$ and $YH^k$ are…

Let $\overline{\mathcal{P}^\star}$ be the family of 3-connected maximal 1-planar graphs, and let $m(\overline{\mathcal{P}^\star},n)$ denote the minimum number of edges among graphs…

Non-perfection conjecture. Every optimal $1$-planar graph with at least $10$ vertices is not perfect.

Let $G$ be an optimal 1-planar graph with $n\geq 10$ vertices, and let $h_{\max}(G)$ denote the maximum length of a hole in $G$. Maximum hole-length conjecture. … The formula count…

Bounded-distance crossings conjecture. There is an integer $k$ such that for every pair of crossing edges $uv,xy$ in $G$, we have

1-planar coarse-planarity conjecture. Every quasi-transitive 1-planar graph of bounded degree is quasi-isometric to a planar graph.

Let $G$ be a connected graph, and let … Delta$(G)$-coloring. This conjecture is an equitable analogue of Brooks' theorem and concerns the equitable chromatic number. The paper's ab…

Let $G$ be a graph, and let $K_2$ denote the complete graph with two vertices. A graph is 1-planar if it has a drawing in the plane in which every edge is crossed by at most one ot…

Ringel's conjecture. Every 1-planar graph $G$ is 6-colorable; equivalently,

Let $N$ be a fixed positive integer. A 1-planar graph is a graph that admits a drawing in which each edge is crossed at most once. Consider a 1-planar graph with minimum degree at…

Near-perfect matching conjecture. Every 5-connected 1-planar graph has a matching of size

Let $G$ be a 1-planar graph with maximum degree $\Delta=\Delta(G)$. Zhang and Liu's conjecture. The bound $\Delta\geq 10$ in the known result that $\chi'(G)=\Delta$ should be reduc…

Let $x$ and $y$ be integers with $x\geq 3$ and $y\geq 6x-12$, and let $G$ be a bipartite 1-planar graph whose partite sets have sizes $x$ and $y$. Extremal edge conjecture. The gra…

Bar 1-visibility conjecture. Every 1-planar graph is a bar 1-visible graph.

Let $G$ be a 1-planar graph, meaning that it has a drawing in the plane in which each edge is crossed by at most one other edge. Let $\Delta(G)$ denote its maximum degree, and let…