24 problems

Higher-outerplanarity obstruction conjecture. For any $k\geq 4$, there exists a 3-connected $k$-outerplanar triangulated disc with no completely independent spanning trees.

Let $\mathcal{L}=G_u\cup H_u^{\triangle}$ be a planar Landau diagram, where $G$ is an outerplanar graph, and let $\Delta_{\mathcal{L}}$ denote its LS discriminant. Cluster Factoriz…

Let $G$ be a subcubic outerplanar graph, meaning that $G$ is outerplanar and has maximum degree at most $3$. For each positive integer $t$, let $\tau_t(G)$ denote the smallest numb…

Equitable coloring conjecture. If

Induced maximal outerplane subgraph conjecture.

Borradaile–Le–Sherman-Bennett conjecture.

Let $H$ be an outerplanar graph, and let $H^+$ denote the graph obtained from $H$ by adding an apex vertex adjacent to every vertex of $H$. Apex-outerplanar minor conjecture. Every…

Maximal outerplanar domination-packing conjecture.

Let $G$ be an outerplanar graph, and let $\mu_{\operatorname{int}}(G)$ denote its interval coloring impropriety. Casselgren–Petrosyan's outerplanar graph conjecture. … This would i…

Extremal $k$th-eigenvalue conjecture. If $n=kq+1$, then for fixed $k$ and sufficiently large $n$,

Let $G$ be a directed acyclic outerplanar graph, and let $\operatorname{sn}(G)$ denote its stack number. Heath, Pemmaraju and Trenk's conjecture. The stack number of the class of d…

Improper coloring conjecture. For the classes $\mathcal{P}$ and $\mathcal{O}$, the following equalities hold:

Maximum-spread outerplanar graph conjecture. For $n$ sufficiently large, the unique $n$-vertex outerplanar graph of maximum spread is

Let $f_{\mathcal{OP}}(n,P_6)$ denote the generalized outerplanar Turán number of the path $P_6$ on $6$ vertices, namely the maximum number of copies of $P_6$ in an outerplanar grap…

A crumby coloring is a red-blue vertex coloring in which the blue subgraph has maximum degree at most $1$ and the red subgraph has minimum degree at least $1$ and contains no path…

Odd–even cycle ratio conjecture. Both ratios

Outerplanar graph star chromatic index conjecture. Every such graph satisfies

Let $G$ be an outerplanar graph with maximum degree $\Delta$. The outerplanar star chromatic index conjecture. … This would improve the known bound…

Let $G$ be an outerplane graph. For each connected component $G_i$ of $G$, let $B_{i,1},\dots,B_{i,b_i}$ be its blocks; let $\mathscr{F}(B_{i,j})$ denote the bounded faces of…

Let $G$ be a subcubic graph, and let $D(G)$ denote its subdivision, obtained by replacing each edge of $G$ by a path of length two. The packing chromatic number $\chi_p(G)$ is the…

Let $G$ be a bi-connected outerplanar graph. Let $G^*$ be its inner dual, and let $P_1\cup\cdots\cup P_t$ be the polygon chain decomposition of $G$ corresponding to the maximal pat…

Let $G$ be a planar graph on $n$ vertices, and let an induced outerplane graph be an induced subgraph whose embedding inherited from $G$ is an outerplanar embedding. Planar outerpl…

A $2$-coloring assigns each vertex a red usage in $[0,1]$, with blue usage $1-r(v)$; the defect of a vertex is the sum, over its neighbors, of the overlap in their color usages, an…

Fix $k$ and consider $k$-outerplanar graphs, with the subclass of 3-connected $k$-outerplanar graphs. A graph property is recognizable if it is recognized by a finite-state tree au…