BeyondPlanarity: Tur\'anType Results for NonPlanar Bipartite Graphs
I S A A C
BeyondPlanarity: Tur?nType Results for NonPlanar Bipartite Graphs
Universit?t T?bingen 0 1 2 3
Germany 0 1 2 3
Michael A. Bekos 0 1 2 3
0 Torsten Ueckerdt Fakulta?t fu?r Informatik, KIT , Karlsruhe , Germany
1 WilhelmSchickhardInstitut fu?r Informatik, Universita?t Tu?bingen , Germany
2 Maximilian Pfister WilhelmSchickhardInstitut fu?r Informatik, Universita?t Tu?bingen , Germany
3 Michael Kaufmann WilhelmSchickhardInstitut fu?r Informatik, Universita?t Tu?bingen , Germany
Beyondplanarity focuses on the study of geometric and topological graphs that are in some sense nearly planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter due to edge crossings should be minimized (e.g., Huang et al. 2018). Most of the literature focuses on Tur?ntype problems, which ask for the maximum number of edges a beyondplanar graph can have. Here, we study this problem for bipartite topological graphs, considering several types of beyondplanar graphs, i.e. 1planar, 2planar, fanplanar, and RAC graphs. We prove bounds on the number of edges that are tight up to additive constants; some of them are surprising and not along the lines of the known results for nonbipartite graphs. Our findings lead to an improvement of the leading constant of the wellknown Crossing Lemma for bipartite graphs, as well as to a number of interesting questions on topological graphs. 2012 ACM Subject Classification Theory of computation ? Computational geometry, Mathematics of computing ? Graph theory
and phrases Bipartite topological graphs; beyond planarity; density; Crossing Lemma

Planarity has been a central concept in the areas of graph algorithms, computational geometry,
and graph theory since the beginning of the previous century. While planar graphs were
originally defined in terms of their geometric representation, they exhibit a number of
combinatorial properties that only depend on their abstract representations. To mention
some of the most important landmarks, we refer to the characterization of planar graphs in
(a) 1planar
(b) 3quasiplanar
(c) fanplanar
(d) RAC
terms of forbidden minors, to the existence of lineartime algorithms to test planarity, to
the FourColor theorem, and to the Euler?s polyhedron formula, which implies that nvertex
planar graphs have at most 3n ? 6 edges.
For the applicative purpose of visualizing realworld networks, however, the concept of
planarity turns out to be overly restrictive. Graphs representing such networks are too dense
to be planar, even though one can often confine nonplanarity in some local structures. Also,
cognitive experiments [28] show that this does not affect the readability of the drawing too
much, if these local structures satisfy specific properties. In other words, these experiments
indicate that even nonplanar drawings may be effective for human understanding, as long as
the crossing configurations satisfy certain properties. Different requirements on the crossing
configurations naturally give rise to different classes of beyondplanar graphs.
Beyondplanarity is then defined as a generalization of planarity, which encompasses all these classes.
Early works date back to 60?s [12] in the field of extremal graph theory, and continued over
the years [3, 7, 33]; also due to the aforementioned experiments, a strong attention on the
topic was recently raised (e.g., [21]), which led to many results, described below.
Some of the most studied beyondplanar graphs include:
(i) kplanar graphs, in which each edge is crossed at most k times [2, 32, 33], see Fig. 1a;
(ii) kquasiplanar graphs, which disallow sets of k pairwise crossing edges [1, 3, 24], see
Fig. 1b;
(iii) fanplanar graphs, in which no edge is crossed by two independent edges or by two
adjacent edges from different directions [14, 30], see Fig. 1c;
(iv) RAC graphs, in which crossings happen at right angles [20, 22]; see Fig. 1d.
Two notable subfamilies of 1planar graphs are the ICplanar graphs [38], where crossings are
independent (i.e., no two crossed edges share an endpoint), and the NICplanar graphs [37],
where crossings are nearly independent (i.e., no two pairs of crossed edges share two endpoints).
For a survey providing an overview on beyondplanarity see [21].
From the combinatorial point of view, the main extremal graph theory question, also
called Tur?ntype [15], concerns the maximum number of edges for graphs in a certain class.
Tight density bounds are known for several classes [20, 30, 33, 37, 38]; a main open question
is to determine the density of kquasiplanar graphs, which is conjectured to be linear in n
for any fixed k [1, 3, 7, 24]. The new bounds for 1, 2, 3 and 4planar graphs have led
to progressive improvements on the leading constant of the lower bound on the number of
1
crossings of a graph, provided by the wellknown Crossing Lemma, from 100 = 0.01 [5, 31]
to 614 ? 0.0156 [4], to 33.75 ? 0.0296 [33], to 31.1 ? 0.0322 [32], to 219 ? 0.0345 [2]. Related
1 1
combinatorial problems concern inclusion relationships between classes [8, 14, 17, 22, 25].
From the complexity side, in contrast to efficient planarity testing algorithms [27],
recognizing a beyondplanar graph has often been proven to be NPhard [10, 14].
Polynomialtime testing algorithms can be found when posing additional restrictions on the produced
drawings, namely, that the vertices are required to lie either on two parallel lines (see,
e.g., [14, 19]) or on the outer face of the drawing (see, e.g., [11, 26]).
Another natural restriction, yet rarely explored in the literature, is to pose additional
structural constraints on the graphs themselves, rather than on their drawings. For
3connected 1plane graphs, Alam et al. [6] presented a polynomialtime algorithm to construct
1planar straightline drawings. Further, Brandenburg [16] gave an efficient algorithm to
recognize optimal 1planar graphs, i.e., those with the maximum number of edges.
For the important class of bipartite graphs, very few results have been discovered so far.
From the density point of view, the only result we are aware of is a tight bound of 3n ? 8
edges for bipartite 1planar graphs [18, 29]. Didimo et al. [19] characterize the complete
bipartite graphs that admit RAC drawings, but their result does not extend to noncomplete
graphs.
Our contribution. Along this direction, we study several classes of beyondplanar bipartite
topological or geometric graphs, focusing on Tur?ntype problems. Table 1 shows our findings.
The new bound on the edge density of bipartite 2planar graphs leads to an improvement
of the leading constant of the Crossing Lemma for bipartite graphs from 219 ? 0.0345,
1
which holds for general graphs [2], to 18.1 ? 0.0554 (see Theorem 16). To the best of our
knowledge, this is the first nontrivial adjustment of the Crossing Lemma that is specific
for bipartite graphs, besides the Zarankiewicz conjecture [36], which however only concerns
complete bipartite graphs. Our results also unveil an interesting tendency in the density of
kplanar bipartite graphs with respect to the one of general kplanar graphs. At first sight,
the differences seem to be around n, as it is in the planar and in the 1planar cases (i.e.,
n ? 2). This turns out to be true also for RAC and fanplanar graphs. However, for IC and
NICplanar graphs, and in particular for 2planar graphs, the differences are larger.
Another notable observation from our results is that, in the bipartite setting, fanplanar
graphs can be denser than 2planar graphs, while in the nonbipartite case these two classes
have the same maximum density, even though none of them is contained in the other [14].
In Section 6 we discuss a number of open problems that are raised by our work.
Methodology. We focus on five classes of bipartite beyondplanar graphs; see Sections 2?5.
To estimate the maximum edge density of each class we employ different counting techniques.
For the class of bipartite ICplanar graphs, we apply a direct counting argument based on
the number of crossings that are possible due to the restrictions posed by ICplanarity.
Our approach is different for NICplanarity. We show that a bipartite NICplanar graph
of maximum density contains a set of uncrossed edges forming a plane subgraph whose
faces have length 6, and that each such face contains exactly one crossing pair of edges.
To estimate the maximum number of edges of a bipartite RAC graph, we adjust a technique
by Didimo et al. [20], who proved the corresponding bound for general RAC graphs.
For fanplanarity, our technique is more involved. After examining structural properties
of maximal bipartite fanplanar graphs, we show how to augment them so that they
contain a planar quadrangulation as a subgraph. Then, we develop a charging scheme
which charges edges involved in fan crossings to the corresponding vertices, to prove that
there are at least as many edges in the quadrangulation as in the rest of the graph.
For 2planarity, we again show that maximal bipartite 2planar graphs have a planar
quadrangulation as a subgraph. We then use a counting scheme based on an auxiliary
directed plane graph, defined by orienting the dual of the quadrangulation, describing
dependencies of adjacent quadrangular faces posed by the edges not belonging to it.
Preliminaries. We consider connected topological graphs, i.e., drawn in the plane with
vertices represented by points in R2 and edges by Jordan arcs connecting their endvertices,
so that:
(i) no edge passes through a vertex different from its endpoints,
(ii) no two adjacent edges cross,
(iii) no edge crosses itself,
(iv) no two edges meet tangentially, and
(v) no two edges cross more than once.
A graph has no selfloops or multiedges. Otherwise, it is a topological multigraph, for which
we assume that the two regions defined by selfloops or multiedges contain at least one vertex
in their interiors, i.e., all edges are nonhomotopic.
We refer to a beyondplanar graph G with n vertices and maximum possible number of
edges as optimal. Consider all the plane spanning subgraphs of G (i.e., in their drawings
inherited from G there exists no two crossing edges). Among those, we select one with the
largest number of edges, which we denote by Gp and call it the planar structure of G. Let
f = hu0, u1, . . . , uk?1i be a face of Gp. We say that f is simple if ui 6= uj for each i 6= j; face
f is connected if edge (ui, ui+1) exists for each i = 0, . . . , k ? 1 (indices modulo k).
2
Bipartite IC and NICPlanar Graphs
In this section, we give tight bounds on the density of bipartite IC and NICplanar graphs.
For the proofs of the lower bounds, we refer to Fig. 2. Full proofs can be found in [9].
I Theorem 1. For infinitely many values of n, there exists a bipartite nvertex ICplanar
graph with 2.25n ? 4 edges, and a bipartite nvertex NICplanar graph with 2.5n ? 5 edges.
I Theorem 2. A bipartite nvertex ICplanar graph has at most 2.25n ? 4 edges
Proof. Our proof is an adjustment of the one for general ICplanar graphs [38]. Let G be
a bipartite nvertex optimal ICplanar graph. Let cr(G) be the number of crossings of G.
Since every vertex of G is incident to at most one crossing, cr(G) ? n4 . By removing one edge
Gk 1
from every pair of crossing edges of G, we obtain a plane bipartite graph, which has at most
2n ? 4 edges. Hence, the number of edges of G is at most 2n ? 4 + cr(G) = 2.25n ? 4. J
I Theorem 3. A bipartite nvertex NICplanar graph has at most 2.5n ? 5 edges.
Proof. Among all bipartite optimal NICplanar graphs with n vertices, let G be the one with
the maximum number of uncrossed edges, i.e., G is such that the plane (bipartite) subgraph
H obtained by removing every crossed edge in G has maximum density. It is not difficult to
show that each face of H containing two crossing edges in G is connected and has length
6 (for details see [9]). Thus, every face of H has length either 6, if it contains two edges
crossing in G, or 4 otherwise (due to bipartiteness and maximality).
Let ? and ? be the number of vertices and edges of H, respectively. Clearly, n = ?. Let
also ?4 and ?6 be the number of faces of length 4 and 6 in H, respectively. We have that
2?4 + 3?6 = ?. By Euler?s formula, we also have that ? + 2 = ? + ?4 + ?6. Combining
these two equations, we obtain: ?4 + 2?6 = ? ? 2. So, in total the number of edges of G is
? + 2?6 = 2?4 + 5?6 = 2n ? 4 + ?6. By Euler?s formula, the number of faces of length 6 of a
planar graph is at most (n ? 2)/2, which implies that G has at most 2.5n ? 5 edges. J
3
Bipartite RAC Graphs
We continue our study on bipartite beyondplanarity with the class of geometric RAC graphs.
We prove an upper bound on their density that is optimal up to a constant of 2.
I Theorem 4. For infinitely many values of n, there exists a bipartite nvertex RAC graph
with 3n ? 9 edges.
Proof. For any k > 1, we recursively define a graph Gk by attaching six vertices and 18
edges to Gk?1; see the left part of Fig. 3. The base graph G1 is a hexagon containing two
crossing edges. So, Gk has 6k vertices and 18k ? 10 edges. The right part of Fig. 3 shows
that Gk is RAC: if Gk?1 is drawn so that its outerface is a parallelogram, then it can be
augmented to a RAC drawing of Gk whose outerface is a parallelogram with sides parallel to
the ones of Gk?1. The bound follows by adding an edge in the outerface of Gk by slightly
?adjusting? its drawing; ses [9]. J
I Theorem 5. A bipartite nvertex RAC graph has at most 3n ? 7 edges.
Proof. Let G be a (possibly nonbipartite) RAC graph with n vertices. Since G does not
contain three mutually crossing edges, as in [20] we can color its edges with three colors (r,
b, g) so that the crossingfree edges are the redges, while bedges cross only gedges, and
viceversa. Thus, the subgraphs Grb, consisting of only r and bedges, and Grg, consisting
of only r and gedges, are both planar. Didimo et al. [20, Lemma 4] showed that each face
of Grb has at least two redges, by observing that if this property did not hold, then the
drawing could be augmented by adding redges. Thus, the number mb of bedges is at most
n ? 1 ? d?/2e, where ? ? 3 is the number of edges in the outer face of G. Suppose now
that G is additionally bipartite. We still have mb ? n ? 1 ? d?/2e, but in this case ? ? 4
holds (by bipartiteness). Hence, mb ? n ? 3. Since Grg is bipartite and planar, it has at
most 2n ? 4 edges (i.e., mr + mg ? 2n ? 4). Hence, G has at most 3n ? 7 edges. J
4
Bipartite FanPlanar Graphs
We continue our study with the class of fanplanar graphs. We begin as usual with the lower
bound (Theorem 6), which we suspect to be bestpossible both for graphs and multigraphs.
For fanplanar bipartite graphs, we prove an almost tight upper bound (Theorem 11).
I Theorem 6. For infinitely many values of n, there exists a bipartite nvertex fanplanar
(i) graph with 4n ? 16 edges, and
(ii) multigraph with 4n ? 12 edges.
Proof sketch. Figs. 4a, 4b, and 4c show constructions that yield bipartite nvertex fanplanar
multigraphs with 4n ? 12 edges. Removing the thick edges in Figs. 4a and 4c gives bipartite
nvertex fanplanar graphs with 4n ? 16 edges. J
To prove the upper bound, consider a bipartite fanplanar graph G with a fixed fanplanar
drawing. W.l.o.g. assume that G is edgemaximal and connected, and A, B are the two
bipartitions of G. We shall denote vertices in A by a, a0, or ai for some index i, and similarly
vertices in B by b, b0, or bi. By fanplanarity, for each crossed edge e of G all edges crossing
e have a common endpoint v. We call e an Aedge (respectively, Bedge) if this vertex v lies
in A (respectively, B). If e is crossed exactly once, it is Aedge and Bedge.
A cell of some subgraph H of G is a connected component c of the plane after removing
all vertices and edges in H; see also [30]. The size of c, denoted by c is the total number
of vertices and edge segments on the boundary ?c of c, counted with multiplicities.
I Lemma 7 ([30]). Each fanplanar graph G admits a fanplanar drawing such that if c is a
cell of any subgraph of G, and c = 4, then c contains no vertex of G in its interior.
We choose a fanplanar drawing of G with the property given in Lemma 7.
I Corollary 8. If e = (a, b), with a ? A and b ? B, is crossed at point p by an Aedge e0,
then each edge crossing e between a and p is an Aedge crossed by each edge crossing e0.
Proof. Let x be the common endpoint of all edges crossing e and e0 = (x, y) be the Aedge
crossing e in p. Let e00 = (x, y0) be an edge that crosses e between p and a. If e00 is not an
Aedge, it is crossed by an edge e1 = (a0, b) with a0 6= a. The Aedge e0 is not crossed by e1.
But then there is a cell c1 bounded by vertex b and segments of e, e00 and e1, which contains
vertex x or y in its interior (see Fig. 5a), contradicting Lemma 7. Symmetrically, if there is
an edge e2 = (a, b0) that crosses e0 but not e00, then there is a similar cell c2 with c2 = 4
containing vertex x or y0 (see Fig. 5b), again contradicting Lemma 7. J
Kaufmann and Ueckerdt [30] derive Lemma 7 from the following lemma.
I Lemma 9 ([30]). Let G be given with a fanplanar drawing. If two edges (v, w) and (u, x)
cross in a point p, no edge at v crosses (u, x) between p and u, and no edge at x crosses
(v, w) between p and w, then u and w are on the boundary of the same cell of G.
By the maximality of G we have in this case that (u, w) is an edge of G, provided u and w
lie in distinct bipartition classes. We can use this fact to derive the following lemma.
I Lemma 10. Assume that e1 = (a1, b1) and e2 = (a2, b2) cross. If e1 and e2 are both A or
Bedges, then (a2, b1) belongs to G and can be drawn so that each edge that crosses (a2, b1)
also crosses e2. Otherwise, (a2, b1) belongs to G and can be drawn crossingfree.
Proof. First assume that e1 and e2 are both Aedges; the case where e1 and e2 are both
Bedges is analogous. Let p1 be the crossing point on e1 that is closest to b1. Since e1 is
an Aedge crossing (a2, b2), the edge e crossing e1 at p1 (possibly e = e2) is incident to a2.
Now either e = e2 or the subgraph H of G consisting of e, e1 and e2 (and their vertices) has
one bounded cell c of size 4, which by Lemma 7 contains no vertex. In both cases, every
edge of G crossing e between a2 and p1, also crosses e2 and ends at a1 (as e2 is an Aedge
crossing (a1, b1)). Thus, drawing an edge from b1 along e1 to p1 and then along e to a2 does
not violate fanplanarity; see Fig. 5c. By the maximality of G, (a2, b1) belongs to G .
Now assume that e1 is an Aedge and e2 is a Bedge. Let p be the crossing point of e1
and e2. By Lemma 9, a2 and b1 lie on the same cell in G and hence, by the maximality of G,
we have that the edge (a2, b1) is contained in G and can be drawn crossingfree. J
We are now ready to prove the main theorem of this section (see also [9] for omitted parts).
I Theorem 11. Any nvertex bipartite fanplanar graph has at most 4n ? 12 edges.
Proof sketch. We start by considering the planar structure Gp of G, i.e., an
inclusionmaximal subgraph of G whose drawing inherited from G is crossingfree. Let EA and EB be
the set of all Aedges and Bedges, respectively, in E[G] ? E[Gp]. Each e ? EA is crossed by
a nonempty (by maximality of Gp) set of edges in G with common endpoint a ? A, and we
say that e charges a. Similarly, every e ? EB charges a unique vertex b ? B.
For any vertex v in G, let ch(v) denote the number of edges in EA ? EB charging v.
Moreover, for a multigraph H containing v, let degH (v) denote the degree of v in H, i.e., the
number of edges of H incident to v. Our goal is to show that for every vertex v of G we have
degGp (v) ? ch(v) ? 2. However, this is not necessarily true when Gp is not connected or has
faces of length 6 or more. To overcome this issue, we shall add in a stepbystep procedure
vertices and edges (possibly parallel but nonhomotopic to existing edges in Gp) to the plane
drawing of Gp such that:
a s
e00
(a)
e
q
b
(P.1) the obtained multigraph G?p is a planar quadrangulation,
(P.2) the drawing of the multigraph G? := G ? G?p is again fanplanar, and
(P.3) each new vertex is added with three edges to other (possibly earlier added) vertices.
To find G?p (refer to [9] for a full proof), we first assume that Gp is not connected. In this
case there must be an edge e with endpoints in different connected components of Gp, which
is crossed by some edge e0 in Gp. Depending on which of e, e0 is an Aedge or Bedge, we
either use Lemma 10 to add a new edge to Gp or we carefully add a new vertex of degree
three to Gp. Once we may assume that Gp is connected but not a quadrangulation, there
exists a face f whose facial walk W has length at least 6. For edges e with one endpoint in
V [W ] that run through face f and leave f by crossing an edge e0 of Gp, we define a stick to
be the initial segment of e that is contained in f . Such a stick s splits W into two parts,
each going from the start vertex of e to the crossing of e and e0. As G is bipartite, exactly
one part, the inner side of s, contains an even number of vertices, and s is called short if its
inner side has only two vertices, and long otherwise. In case f has a long stick, then again
depending on which of e, e0 is an Aedge or Bedge, we either use Lemma 10 to add a new
edge to Gp (see Fig. 6a) or we carefully add a new vertex of degree three to Gp (see Fig. 6b).
Finally, if all sticks are short, we can add a crossingfree edge to Gp, or a new vertex with
three crossingfree edges to Gp, as shown in Fig. 6c.
Adding to Gp an edge or a vertex with three edges, strictly increases the average degree
in Gp. Hence, we ultimately obtain supergraphs G? of G and G?p of Gp satisfying P.1?P.3.
Next, we show that the charge of every original vertex v is at most its degree in G?p minus 2.
I Claim 12. Every v ? V [G] satisfies degG?p (v) ? ch(v) ? 2.
Proof. W.l.o.g. consider any a ? A and let k := degG?p (a) and S ? EA be the set of edges
charging a. Observe that no two edges of S can cross. In fact, if (a1, b1) ? EA charges a and
(a2, b2) ? EA crosses (a1, b1), then (a2, b2) charges a1 6= a. Consider the face f of G?p ? {a}
containing a, and the closed facial walk W around f . Walk W has length 2k (counting with
repetitions) as G?p is a quadrangulation. Further, each edge in S lies in f and has both
endpoints on W . Hence, the subgraph of G? consisting of all edges in W ? S is crossingfree
and has vertex set V [W ]. Define graph J by breaking the repetitions along W , i.e., J consists
of a cycle of length 2k and every edge in S is an uncrossed chord of this cycle. J has ? k ? 2
chords, as it is bipartite outerplanar. Thus, S = ch(a) ? k ? 2 = degG?p (a) ? 2. J
Let X = V [G?] ? V [G] be the set of newly added vertices. For each x ? X, degG?p (x) ? 3 and
ch(x) = 0 hold. Thus, degG?p (x)?ch(x) ? 3, and by Claim 12 we get 2E[G?p]?(EA+EB) =
Pv?V [G?p] degG?p (v) ? ch(v) ? 2n + 3X which implies EA + EB ? 2E[G?p] ? 2n ? 3X.
On the other hand, E[Gp] + 3X ? E[G?p] by P.3 and E[G?p] = 2(n + X) ? 4 by P.1,
which together give E[G] = E[Gp] + EA + EB ? 3E[G?p] ? 6X ? 2n = 4n ? 1. J
(a)
ui+2
s
ui ui+1
s
In this section, we overview our result for bipartite 2planar graphs. For reasons of space, we
sketch the proof; the full version is in [9]. We start with the lower bound; see Fig.7.
I Theorem 13. For infinitely many values of n, there exists a bipartite nvertex 2planar
(i) graph with 3.5n ? 12 edges, and
(ii) multigraph with 3.5n ? 8 edges.
For the upper bound, we study structural properties of the planar structure Gp of an
optimal bipartite 2planar graph G. Let (u, v) be an edge of G that does not belong to Gp.
By the maximality of Gp, (u, v) has at least one crossing with an edge of Gp. As already
mentioned, the part of (u, v) that starts from u (v) and ends at the first intersection point of
(u, v) with an edge of Gp is a stick of u (v). When (u, v) has two crossings, there is a part
that is not a stick, called middlepart. Each stick or middlepart lies in a face f of Gp; we say
that f contains this part. Let f = hu0, u1, . . . , uk?1i be a face of Gp with k ? 4 and let s be
a stick of ui contained in f , i ? {0, 1, . . . , k ? 1}. We call s a short stick, if it ends either at
(ui+1, ui+2) or at (ui?1, ui?2) of f ; otherwise, s is called a long stick; see Figs. 8a8b.
W.l.o.g. we assume that among all optimal bipartite nvertex 2planar graphs, G is such
that its planar structure Gp is the densest among the planar structures of all other optimal
bipartite nvertex 2planar graphs; we call Gp maximally dense. We first prove that Gp is a
spanning quadrangulation. For this, we first show that Gp is connected, as otherwise it is
always possible to augment it by adding an edge joining two connected components of it.
Then, we show that all faces of Gp are of length four. Our proof by contradiction is rather
technical; assuming that there is a face f with length greater than four in Gp, we consider
two main cases:
(i) f contains no sticks, but middleparts, and
(ii) f contains at least one stick.
With a careful case analysis, we lead to a contradiction either to the maximality of Gp or to
the fact that G is optimal.
Since Gp is a quadrangulation, it has exactly 2n ? 4 edges and n ? 2 faces. Our goal is to
prove that the average number of sticks for a face is at most 3. Since the number of edges of
G \ Gp equals half the number of sticks over all faces of Gp, this implies that G cannot have
more than 2n ? 4 + 32 (n ? 2) = 3.5n ? 7 edges, which gives the desired upper bound.
f1
(a)
f
f2
f1
(b)
f
f2
(c)
Let f be a face of Gp. Denote by h(f ) the number of sticks contained in f . A scissor of f
is a pair of crossing sticks starting from nonadjacent vertices of f , while a twin of f is a pair
of sticks starting from the same vertex of f crossing the same boundary edge of f ; see Fig. 8c.
We refer to a pair of crossing sticks starting from adjacent vertices of f as a pseudoscissor ;
see Fig. 8d. The following lemma shows that a face of Gp contains a maximum number of
sticks (that is, 4) only in the presence of scissors or twins, due to 2planarity; see [9].
I Lemma 14. Let G be an optimal bipartite 2planar graph, such that its planar structure
Gp is maximally dense. Then, for each face f of Gp, it holds h(f ) ? 4. Further, if h(f ) = 4,
then f contains one of the following: two scissors, or two twins, or a scissor and a twin.
An immediate consequence of Lemma 14 is that h(f ) ? 3, for every face f containing a
pseudoscissor. We now consider specific ?neighboring? faces of a face f of Gp with four sticks
and prove that they cannot contain so many sticks. Observe that each edge corresponding to
a stick of f starts from a vertex of f and ends at a vertex of another face of Gp. We call this
other face, a neighbor of this stick. The set of neighbors of the sticks forming a scissor (twin)
of f form the socalled neighbors of this scissor (twin).
By Lemma 14 and since h(f ) = 4, face f contains two sticks s1 and s2 forming a twin or
a scissor, with neighbors f1 and f2. By 2planarity and based on a technical case analysis,
we show that h(f1) + h(f2) ? 7 except for a single case, called 8sticks configuration and
illustrated in Fig. 9a, for which h(f1) + h(f2) = 8.
Assume first that G does not contain any 8sticks configuration. Let H be an auxiliary
graph, called dependency graph, having a vertex for each face of Gp. Then, for each face f of
Gp containing a scissor or a twin with neighbors f1 and f2, s.t. h(f1) ? h(f2), there is an
edge from f to f1 in H; f1 = f2 is possible. To prove that the average number of sticks for a
face of Gp is at most 3 (which implies the upper bound), it suffices to prove that the number
of faces of Gp that contain two sticks is at least as large as the number of faces that contain
four sticks. This holds due to the following facts for every face f of Gp:
(i) if h(f ) = 4, then f has two outgoing edges and no incoming edge in H,
(ii) if h(f ) = 3, then the number of outgoing edges of f in H is at least as large as the
number of its incoming edges, and
(iii) if h(f ) = 2, then f has at most two incoming edges in H.
So, G has at most 3.5n ? 7 edges in the absence of 8sticks configurations.
Finally, if G contains 8sticks configurations, we eliminate each of them (without creating
new) by adding one vertex, and by replacing two edges of G by six other edges violating
neither bipartiteness nor 2planarity, as in Fig. 9b. The derived graph G0 has a planar
structure that is a spanning quadrangulation without 8sticks configurations. Since G0 has
one vertex and four edges more than G for each 8sticks configuration and since the vertices
of G0 have degree at most 3.5 on average, by reversing the augmentation steps we conclude
that G cannot be denser than G0. We summarize our result in the following.
I Theorem 15. A bipartite nvertex 2planar multigraph has at most 3.5n ? 7 edges.
Implications of Theorem 15. In the following, we adjust the wellknown Crossing Lemma
to bipartite graphs and use it to obtain a bound on the density of bipartite kplanar graphs,
when k > 2. Our proofs are inspired by the ones for general graphs; see, e.g., [4].
I Theorem 16. Let G be a bipartite topological graph with n ? 3 vertices and m ? 147 n
edges. Then, cr(G) ? 289 ? n2 ? 18.1 ? mn23 , where cr(G) is the crossing number of G.
16 m3 1
Proof. We first prove a weaker bound which holds for every m, that is, cr(G) ? 3m? 127 n+19.
This bound clearly holds when m ? 2n ? 4. Hence, we may assume w.l.o.g. that m > 2n ? 4.
It follows from [18] that if m > 3n ? 8, then G has an edge that is crossed by at least two
other edges. Also, by Theorem 15 we know that if m > 72 n ? 7, then G has an edge that is
crossed by at least three other edges. We obtain by induction on the number of edges of G
that cr(G) ? (m ? (2n ? 4)) + (m ? (3n ? 8)) + (m ? ( 7 n ? 7)) = 3m ? 127 n + 19.
2
Assume that G admits a drawing on the plane with cr(G) crossings and let p = 147mn ? 1.
Choose independently every vertex of G with probability p, and denote by Hp the graph
induced by the chosen vertices. Let also np, mp and cp be the random variables corresponding
to the number of vertices, of edges and of crossings of Hp. Taking expectations on the
relationship cp ? 3mp ? 127 np + 19, which holds by our weaker bound, we obtain that
p4cr(G) ? 3p2m ? 127 np, or equivalently that cr(G) ? p2 ? 127pn3 . The proof follows by
3m
plugging p = 147mn (which is at most 1 by our assumption) to the last inequality. J
I Theorem 17. Let G be a bipartite kplanar graph with n ? 3 vertices and m edges, for
17 ?2kn ? 3.005?kn.
some k ? 1. Then: m ? 8
Proof. For k = 1, 2, the bounds are weaker than the ones of [18] and of Theorem 15. So, we
may assume w.l.o.g. that k > 2. We may also assume that m ? 147 n, as otherwise there is
nothing to prove. Combining the fact that G is kplanar with the bound of Theorem 16 we
obtain that 21869 ? mn23 ? cr(G) ? 21 mk, which implies that m ? 187 ?2kn ? 3.005?kn. J
6
Conclusions and Open Problems
We presented tight bounds for the density of bipartite beyondplanar graphs, yielding an
improvement of the leading constant of the Crossing Lemma for bipartite graphs. We
conclude with open problems.
(i) What is the maximum density of bipartite kplanar graphs with k > 2? Such bounds
may further improve the leading constant of the Crossing Lemma for bipartite graphs;
Fig. 9c shows a bipartite 3planar graph with 4n ? O(1) edges. Bounds for other classes
of bipartite beyondplanar (e.g., quasiplanar) graphs are also interesting.
(ii) The ratio of the maximum density of general over bipartite graphs for large n approaches
32nn = 1.5 for planar graphs, 43nn ? 1.33 for 1planar graphs, 35.5nn ? 1.43 for 2planar
graphs and at most 54.5nn ? 1.37 for 3planar graphs, leaving room for speculation
on how it develops for kplanar graphs with k > 3; note that for classes closed under
subgraphs, it is at most 2 [23].
(iii) Optimal 1, 2 and 3planar graphs allow for characterizations [13, 35], while recognizing
general beyondplanar graphs is often NPhard. Does the restriction of bipartiteness
allow for characterizations or efficient recognition algorithms in some cases?
(iv) Finally, one should study properties that not only hold for general beyondplanar graphs
but also for bipartite ones, e.g., is every optimal bipartite RAC graph also 1planar?
1
2
3
Eyal Ackerman . On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges . Discrete Comput. Geom. , 41 ( 3 ): 365  375 , 2009 . doi: 10 .1007/ s0045400991439.
Eyal Ackerman . On topological graphs with at most four crossings per edge . CoRR, abs/1509 . 01932 , 2015 . arXiv: 1509 . 01932 .
Pankaj K. Agarwal , Boris Aronov, J?nos Pach, Richard Pollack, and Micha Sharir. QuasiPlanar Graphs Have a Linear Number of Edges. Combinatorica , 17 ( 1 ): 1  9 , 1997 . doi: 10 .1007/BF01196127.
Martin Aigner and G?nter M. Ziegler . Proofs from THE BOOK (3rd . ed.). Springer, 2004 .
M. Ajtai , V. Chv?tal , M. Newborn , and E. Szemer?di . Crossingfree subgraphs . Annals of Discrete Mathematics , 12 : 9  12 , 1982 .
Jawaherul Alam , Franz J. Brandenburg , and Stephen G. Kobourov. StraightLine Grid Drawings of 3Connected 1 Planar Graphs . In Graph Drawing , volume 8242 of LNCS , pages 83  94 . Springer, 2013 .
Noga Alon and Paul Erd?s . Disjoint Edges in Geometric Graphs. Discrete Comput. Geom. , 4 : 287  290 , 1989 . doi: 10 .1007/BF02187731.
Patrizio Angelini , Michael A. Bekos , Franz J. Brandenburg , Giordano Da Lozzo, Giuseppe Di Battista, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Ignaz Rutter . On the Relationship between kPlanar and kQuasi Planar Graphs . In WG , volume 10520 of LNCS , pages 59  74 . Springer, 2017 . doi: 10 .1007/9783 319 687056\_5.
Patrizio Angelini , Michael A. Bekos , Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt . BeyondPlanarity: Density Results for Bipartite Graphs . CoRR, abs/1712.09855, 2017 . arXiv: 1712 . 09855 .
Evmorfia N. Argyriou , Michael A. Bekos , and Antonios Symvonis . The StraightLine RAC Drawing Problem is NPHard . J. Graph Algor. Appl. , 16 ( 2 ): 569  597 , 2012 . doi: 10 .7155/ jgaa.00274.
Christopher Auer , Christian Bachmaier, Franz J. Brandenburg , Andreas Glei?ner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber . Outer 1 Planar Graphs . Algorithmica, 74 ( 4 ): 1293  1320 , 2016 .
S. Avital and H. Hanani . Graphs. Gilyonot Lematematika , 3 : 2  8 , 1966 .
Michael A. Bekos , Michael Kaufmann, and Chrysanthi N. Raftopoulou . On the Density of Nonsimple 3Planar Graphs . In Graph Drawing , volume 9801 of LNCS , pages 344  356 .
Springer , 2016 . doi: 10 .1007/9783 319 501062\_ 27 .
Theor . Comp. Sci., 589 : 76  86 , 2015 .
B. Bollob?s . Combinatorics: Set Systems , Hypergraphs, Families of Vectors, and Combinatorial Probability . Cambridge University Press, 1986 .
Franz J. Brandenburg . Recognizing Optimal 1 Planar Graphs in Linear Time . Algorithmica, 80 ( 1 ): 1  28 , 2018 . doi: 10 .1007/s0045301602268.
Comp. Sci., 636 : 1  16 , 2016 . doi: 10 .1016/j.tcs. 2016 . 04 .026.
J?lius Czap , Jakub Przyby?o, and Erika ?krabul'?kov?. On an extremal problem in the class of bipartite 1planar graphs . Discussiones Mathematicae Graph Theory , 36 ( 1 ): 141  151 , 2016 .
Walter Didimo , Peter Eades , and Giuseppe Liotta . A characterization of complete bipartite RAC graphs . Inf. Process. Lett., 110 ( 16 ): 687  691 , 2010 . doi: 10 .1016/j.ipl. 2010 . 05 .
Walter Didimo , Peter Eades , and Giuseppe Liotta . Drawing graphs with right angle crossings . Theor. Comp. Sci. , 412 ( 39 ): 5156  5166 , 2011 .
Walter Didimo , Giuseppe Liotta, and Fabrizio Montecchiani . A Survey on Graph Drawing Beyond Planarity . CoRR, abs/ 1804 .07257, 2018 . arXiv: 1804 .07257.
Peter Eades and Giuseppe Liotta . Right angle crossing graphs and 1planarity . Discrete Appl . Math., 161 ( 78 ): 961  969 , 2013 . doi: 10 .1016/j.dam. 2012 . 11 .019.
Paul Erd?s . On some extremal problems in graph theory . Israel J. Math. , 3 : 113  116 , 1965 .
SIAM J . Discrete Math., 27 ( 1 ): 550  561 , 2013 . doi: 10 .1137/110858586.
Michael Hoffmann and Csaba D. T?th . TwoPlanar Graphs Are Quasiplanar . In MFCS , volume 83 of LIPIcs , pages 47 : 1  47 : 14 . Schloss Dagstuhl , 2017 . doi: 10 .4230/LIPIcs.MFCS.
SeokHee Hong , Peter Eades , Naoki Katoh, Giuseppe Liotta, Pascal Schweitzer, and Yusuke Suzuki . A LinearTime Algorithm for Testing Outer1Planarity . Algorithmica, 72 ( 4 ): 1033  1054 , 2015 .
John E. Hopcroft and Robert Endre Tarjan . Efficient Planarity Testing. J. ACM , 21 ( 4 ): 549  568 , 1974 . doi: 10 .1145/321850.321852.
Weidong Huang , SeokHee Hong , and Peter Eades . Effects of Crossing Angles . In PacificVis 2008 , pages 41  46 . IEEE, 2008 .
D. V. Karpov . An Upper Bound on the Number of Edges in an Almost Planar Bipartite Graph . Journal of Mathematical Sciences , 196 ( 6 ): 737  746 , 2014 .
Michael Kaufmann and Torsten Ueckerdt . The Density of FanPlanar Graphs . CoRR , 1403 .6184, 2014 . arXiv: 1403 . 6184 .
Frank Thomson Leighton . Complexity Issues in VLSI: Optimal Layouts for the Shuffleexchange Graph and Other Networks . MIT Press, Cambridge, MA, USA, 1983 .
J?nos Pach , Rado? Radoi?i?, G?bor Tardos, and G?za T?th . Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs . Discrete Comput. Geom. , 36 ( 4 ): 527  552 , 2006 .
J?nos Pach and G?za T?th . Graphs Drawn with Few Crossings per Edge . Combinatorica, 17 ( 3 ): 427  439 , 1997 . doi: 10 .1007/BF01215922.
Gerhard Ringel . Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamb. , 29 : 107  117 , 1965 .
Yusuke Suzuki . Reembeddings of Maximum 1Planar Graphs . SIAM J. Discrete Math., 24 ( 4 ): 1527  1540 , 2010 . doi: 10 .1137/090746835.
Kazimierz Zarankiewicz . On a Problem of P. Tur?n Concerning Graphs . Fundamenta Mathematicae , 41 : 137  145 , 1954 .
Xin Zhang . Drawing complete multipartite graphs on the plane with restrictions on crossings . Acta Mathematica Sinica , English Series, 30 ( 12 ): 2045  2053 , 2014 .
Xin Zhang and Guizhen Liu. The structure of plane graphs with independent crossings and its applications to coloring problems . Central Eur. J. of Mathematics , 11 ( 2 ): 308  321 , 2013 .