44 problems

Let $G$ be an outerplanar graph on $n$ vertices, let $\lambda(G)$ denote the spectral radius of its adjacency matrix, let $P_{n-1}$ be the path on $n-1$ vertices, and let $\vee$ de…

Equitable coloring conjecture. If

Induced maximal outerplane subgraph conjecture.

Outerplanar graph star chromatic index conjecture. Every such graph satisfies

Bezegová et al.'s conjecture. The star chromatic index of every outerplanar graph satisfies

A crumby coloring of a graph is a red–blue vertex partition in which the blue vertices induce a graph of maximum degree at most one, while the red vertices induce a graph with no i…

Let $G$ be an outerplanar graph of order $n$, and let $s(G)=\lambda_1(G)-\lambda_n(G)$ denote the spread of its adjacency matrix. Let $P_{n-1}$ be the path on $n-1$ vertices, let…

Let $G$ be an outerplanar graph, and let $\operatorname{\mu_{int}}(G)$ denote the smallest integer $k$ for which $G$ has an improper interval edge coloring such that at most $k$ ed…

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

A graph is planar if it can be drawn in the plane without crossings, and it is outerplanar if it has a planar drawing in which every vertex lies on the boundary of the outer face.…

Random outerplanar graph conjecture. With probability $1-o(1)$,

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 $S_{2,3}$ be the double star obtained from an edge by joining one endpoint to two isolated vertices and the other endpoint to three isolated vertices. Let…

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…

Borradaile–Le–Sherman-Bennett conjecture.

Asymptotic B-coloring conjecture. For sufficiently large $\Delta$, if $G$ is planar then

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.

Third-eigenvalue conjecture for order $3q$. For sufficiently large $n$, among connected outerplanar graphs on $n=3q$ vertices, the graph maximizing $\lambda_3$ is unique and isomor…

Even-order extremal second-eigenvalue conjecture. For every even $n\geq14$, the connected outerplanar graph maximizing $\lambda_2$ is unique and isomorphic to…

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…

Let $G$ be a rectangle visibility graph, and let $\operatorname{height}(G)$ denote the minimum height among its rectangle-visibility representations. Height-two rectangle visibilit…

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