1-Disk OX Drawing in Bipartite 1-Planar Graphs

• The paper establishes that 1-disk OX drawings of bipartite 1-planar graphs achieve the sharp bound |E(G)| ≤ 2|V(G)| + |X| - 6, refining previous density estimates.
• 1-Disk OX drawing is a topological model where vertices in set X lie on a disk boundary and vertices in Y are interior, enforcing unique drawing constraints.
• The study employs maximal outerplanar graph constructions and a doubling-and-gluing method to rigorously prove the extremal edge-count bound.

1-Disk OX Drawing, more precisely a $1$-disk $\mathcal O_X$ drawing, is a constrained $1$-planar drawing of a bipartite graph $G$ with partite sets $X$ and $Y$ in which all vertices of $X$ lie on the boundary of a disk $\mathcal O$, while all vertices of $Y$ and all edges lie in the interior of $\mathcal O$, up to a homeomorphism of the plane. The notion was first proposed by Huang, Ouyang, and Dong in connection with the edge density of bipartite $\mathcal O_X$0-planar graphs, and its current extremal theory is centered on the sharp bound $\mathcal O_X$1 for $\mathcal O_X$2 (Wang, 26 Jul 2025).

1. Definition and topological model

A graph is $\mathcal O_X$3-planar if it admits a drawing in the plane such that each edge is crossed at most once. A graph is bipartite if its vertex set can be partitioned into two subsets $\mathcal O_X$4 and $\mathcal O_X$5 such that every edge joins a vertex in $\mathcal O_X$6 and a vertex in $\mathcal O_X$7. In a $\mathcal O_X$8-disk $\mathcal O_X$9 drawing, these two conditions are combined with a topological disk constraint: all vertices of $1$0 lie on the boundary $1$1, all vertices of $1$2 lie strictly in the interior of $1$3, and all edges are drawn as arcs entirely inside $1$4 (Wang, 26 Jul 2025).

This places the class between ordinary bipartite $1$5-planar graphs and more specialized outer-face models. Conceptually, it is a kind of outer-bipartite $1$6-planar drawing: one partite set is forced onto the boundary, while the other partite set and all edge geometry remain internal. The phrase “up to a homeomorphism of the plane” means that the exact Euclidean shape of the disk is irrelevant; only the topological role of a single boundary component matters (Wang, 26 Jul 2025).

The definition is asymmetric in $1$7 and $1$8. The partite set $1$9 is the designated boundary set, and the notation $G$0 records that choice. The proofs exploit this asymmetry heavily, especially the fact that the $G$1-vertices occur along the boundary of a face and can therefore be duplicated and glued in a controlled way on the sphere (Wang, 26 Jul 2025).

2. Sharp extremal bound

The central theorem is exact. If $G$2 is a bipartite graph with partite sets $G$3 and $G$4, where $G$5, and $G$6 has a $G$7-disk $G$8 drawing, then

$G$9

Since $X$0, the same bound can be written as

$X$1

and this upper bound is tight for every feasible pair $X$2 (Wang, 26 Jul 2025).

This result sharply improves the density estimate known for general bipartite $X$3-planar graphs. Huang–Ouyang–Dong proved that every bipartite $X$4-planar graph with partite sets $X$5 and $X$6, $X$7, satisfies

$X$8

and that bound is tight in the unrestricted bipartite $X$9-planar setting. By contrast, the $Y$0-disk $Y$1 constraint reduces the $Y$2-term from $Y$3 to $Y$4 (Wang, 26 Jul 2025).

Placed against other classical bounds, the class is structurally sparse but not planar-sparse. Planar graphs on $Y$5 vertices have at most $Y$6 edges, bipartite planar graphs have at most $Y$7 edges, and general $Y$8-planar graphs have at most $Y$9 edges. Karpov’s bound for bipartite $X$0-planar graphs is $X$1 for even $X$2, and $X$3 for odd $X$4 and for $X$5. For $X$6-disk $X$7 drawings, the sharper formula implies

$X$8

because $X$9 (Wang, 26 Jul 2025). This suggests that the boundary constraint imposes a substantial combinatorial penalty relative to unrestricted bipartite $\mathcal O$0-planarity.

3. Extremal constructions

Tightness is established by an explicit family of drawings. For the base case $\mathcal O$1, the construction starts with a maximal outerplanar graph $\mathcal O$2 on $\mathcal O$3 vertices. Such a graph has exactly $\mathcal O$4 triangular faces. Into each triangular face, one inserts a configuration $\mathcal O$5, then deletes all original edges of $\mathcal O$6. The resulting graph $\mathcal O$7 has $\mathcal O$8 “black” vertices on the boundary and $\mathcal O$9 “blue” vertices in the interior; the blue vertices are independent and each blue vertex is adjacent to three black vertices (Wang, 26 Jul 2025).

The counting is exact. Each triangular face contributes three blue vertices and nine blue edges, so

$Y$0

With $Y$1, this yields

$Y$2

Hence the upper bound is attained in this parameter regime (Wang, 26 Jul 2025).

The general case $Y$3, $Y$4, is obtained by adding $Y$5 further interior vertices into an arbitrary region incident to two black boundary vertices and joining each new vertex to those two black vertices without any edge crossing. This preserves bipartiteness, preserves the $Y$6-disk $Y$7 property, and adds exactly $Y$8 vertices and $Y$9 edges, so equality remains

$\mathcal O$0

for every pair $\mathcal O$1 (Wang, 26 Jul 2025).

The construction is notable for its use of maximal outerplanar graphs as a boundary skeleton. Their triangulated outer-face structure creates $\mathcal O$2 triangular “slots,” each of which hosts one local $\mathcal O$3 gadget. The coefficient pattern in the extremal formula, $\mathcal O$4, is reflected directly in this face-by-face packing mechanism (Wang, 26 Jul 2025).

4. Proof strategy

The upper bound is proved by a doubling-and-gluing argument that reduces the $\mathcal O$5-disk problem to a known theorem on general bipartite $\mathcal O$6-planar graphs. Start with a $\mathcal O$7-disk $\mathcal O$8 drawing $\mathcal O$9 of $\mathcal O_X$00, with all $\mathcal O_X$01-vertices on the boundary and all $\mathcal O_X$02-vertices and edges inside the disk. Perform a spherical mapping to obtain another $\mathcal O_X$03-planar drawing $\mathcal O_X$04 of the same graph in which the $\mathcal O_X$05-vertices still lie on the boundary of $\mathcal O_X$06, but all $\mathcal O_X$07-vertices and edges lie in the exterior of $\mathcal O_X$08. The unbounded face of $\mathcal O_X$09 becomes a bounded face $\mathcal O_X$10 of $\mathcal O_X$11 (Wang, 26 Jul 2025).

Next, place the original drawing $\mathcal O_X$12 into the face $\mathcal O_X$13 of $\mathcal O_X$14 and identify the two boundary copies of $\mathcal O_X$15. The resulting graph $\mathcal O_X$16 is bipartite and $\mathcal O_X$17-planar, with

$\mathcal O_X$18

Applying the Huang–Ouyang–Dong bound

$\mathcal O_X$19

then gives

$\mathcal O_X$20

hence

$\mathcal O_X$21

The proof is short because the geometric constraint is encoded globally by the gluing step rather than by local case analysis (Wang, 26 Jul 2025).

Historically, Huang–Ouyang–Dong had already proved that if $\mathcal O_X$22, then any bipartite graph with a $\mathcal O_X$23-disk $\mathcal O_X$24 drawing satisfies $\mathcal O_X$25, and they asked whether

$\mathcal O_X$26

holds in general. The exact theorem $\mathcal O_X$27 is stronger and resolves that problem completely (Wang, 26 Jul 2025).

5. Relation to other disk-based drawing models

Several nearby graph-drawing models also use disks or a single bounded region, but they are formally different.

Model	Vertex placement	Edge/routing rule
$\mathcal O_X$28-disk $\mathcal O_X$29 drawing	$\mathcal O_X$30 on one disk boundary; $\mathcal O_X$31 inside	Bipartite and $\mathcal O_X$32-planar (Wang, 26 Jul 2025)
Outerplanar strict confluent drawing	All vertices on one disk boundary	Edges are unique smooth paths through arcs and junctions (Eppstein et al., 2013)
Disk arrangement drawing of clustered graphs	Vertices inside cluster disks	Inter-cluster edges routed through pipes (Mchedlidze et al., 2018)
Disk-link drawing	Each vertex is an open disk of radius $\mathcal O_X$33	Each edge is a straight segment between disk centers (Bekos et al., 2020)
Outer-$\mathcal O_X$34-planar orthogonal bar-drawing	All vertices on the outer face as bars	At most two bends per edge, embedding preserved (Biedl, 2020)

Outerplanar strict confluent drawings are the closest single-disk analogue in topological form, but they place all vertices on the boundary and interpret adjacency via unique smooth paths through a track system rather than via ordinary $\mathcal O_X$35-planar arcs. Clustered-graph drawings on disk arrangements use multiple disks and a pipe model for inter-cluster routing. Disk-link drawings replace point vertices by open disks with radius $\mathcal O_X$36 and require each edge to be a straight segment between centers, with no non-incident disk intersected by an edge [(Eppstein et al., 2013); (Mchedlidze et al., 2018); (Bekos et al., 2020)].

A different nearby literature studies outer-$\mathcal O_X$37-planar graphs with all vertices on the outer face. In that setting, planar visibility representations can require $\mathcal O_X$38 area, while orthogonal bar-drawings with crossings and at most two bends per edge can achieve $\mathcal O_X$39 area (Biedl, 2020). This is related in spirit but not equivalent: $\mathcal O_X$40-disk $\mathcal O_X$41 drawings constrain only one partite set to the boundary, not the entire vertex set.

Another separate line studies arrangements of prescribed disks on equally spaced rays from the origin and proves that a greedy strategy yields a covering disk of radius at most $\mathcal O_X$42, with applications to unordered tree drawings with perfect angular resolution (Halupczok et al., 2011). Despite the shared “one-disk” language, that model concerns geometric disk placement and tree embedding, not bipartite $\mathcal O_X$43-planarity.

6. Significance and directions

The significance of the $\mathcal O_X$44-disk $\mathcal O_X$45 model lies in the way a single global topological constraint changes the extremal combinatorics of $\mathcal O_X$46-planar graphs. General bipartite $\mathcal O_X$47-planar graphs allow the tight bound $\mathcal O_X$48, whereas the disk-boundary restriction lowers this to $\mathcal O_X$49 (Wang, 26 Jul 2025). This suggests that forcing one partite set onto a single boundary component imposes a much stronger structural discipline than bipartiteness and $\mathcal O_X$50-planarity alone.

The extremal constructions also indicate why the class remains nontrivial. The boundary vertices of $\mathcal O_X$51 serve as a fixed outer scaffold, while interior vertices in $\mathcal O_X$52 are packed into triangular regions created by a maximal outerplanar skeleton. In the sharp examples, each interior gadget contributes a controlled amount of $\mathcal O_X$53-planar density without violating the single-disk boundary condition (Wang, 26 Jul 2025).

The note solving the extremal problem does not explicitly list new open problems. A plausible continuation is the study of recognition, structural characterization, and extensions in which the disk constraint is modified: for example, multi-disk analogues, $\mathcal O_X$54-planar variants, or more general classes of boundary-constrained bipartite drawings. Another plausible direction is to compare the exact extremal behavior of $\mathcal O_X$55-disk $\mathcal O_X$56 drawings with the algorithmic and geometric constraints known for clustered disk arrangements, disk-link drawings, and outer-$\mathcal O_X$57-planar visibility models (Mchedlidze et al., 2018, Bekos et al., 2020, Biedl, 2020).