A Note on the Maximum Rectilinear Crossing Number of Spiders

Abstract

The maximum rectilinear crossing number of a graph G is the maximum number of crossings in a good straight-line drawing of G in the plane. In a good drawing any two edges intersect in at most one point (counting endpoints), no three edges have an interior point in common, and edges do not contain vertices in their interior. A spider is a subdivision of K1,kK_{1,k}. We provide both upper and lower bounds for the maximum rectilinear crossing number of spiders. While there are not many results on the maximum rectilinear crossing numbers of infinite families of graphs, our methods can be used to find the exact maximum rectilinear crossing number of K1,kK_{1,k} where each edge is subdivided exactly once. This is a first step towards calculating the maximum rectilinear crossing number of arbitrary trees.

Full text

arXiv:1808.00385v1 [math.CO] 1 Aug 2018
A Note on the Maximum Rectilinear Crossing Number of Spiders
Joshua Fallon ∗Kirsten Hogenson †Lauren Keough ‡Mario Lomel´ı §
Marcus Schaefer ¶Pablo Sober´on k
August 2, 2018
Abstract
The maximum rectilinear crossing number of a graph Gis the maximum number of crossings in a
good straight-line drawing of Gin the plane. In a good drawing any two edges intersect in at most one
point (counting endpoints), no three edges have an interior point in common, and edges do not contain
vertices in their interior. A spider is a subdivision of K1,k . We provide both upper and lower bounds for
the maximum rectilinear crossing number of spiders. While there are not many results on the maximum
rectilinear crossing numbers of infinite families of graphs, our methods can be used to find the exact
maximum rectilinear crossing number of K1,k where each edge is subdivided exactly once. This is a first
step towards calculating the maximum rectilinear crossing number of arbitrary trees.
1 Introduction
Planar graphs, which are graphs that can be embedded in the plane such that no two edges intersect except
at their endpoints, have long been of interest to the mathematical community. Kuratowski’s theorem from
1930 gives us a characterization of which graphs are planar [21]. For graphs that are not planar, it is natural
to wonder which drawings in the plane have the fewest crossings. Tur´an did just this during his time in a
labor camp during World War II [29].
Definition. The crossing number of a graph Gis the minimum number of crossings over all drawings of G.
The crossing number for Km,n, originally investigated by Tur´an, is still unknown. The conjecture for the
optimal bound is known as the Zarankiewicz conjecture [33]. Similarly, finding the crossing number of Kn
is open and the conjecture for the optimal bound is known as Hill’s conjecture [19]. The problem of finding
the crossing number of a given graph is NP-complete [17]. For some recent problems on crossing numbers
see Chapter 13 of [18].
There are many variants of the crossing number parameter; for a detailed survey see [27]. One such
variant is the maximum crossing number which aims at maximizing the number of crossings. For this to
make sense, we must insist that we have a good drawing. In a good drawing, no edge can cross itself, each
pair of edges have at most one point in common (including endpoints), and no more than two edges can
cross at any point.
Definition. The maximum crossing number of a graph Gis the maximum number of crossings in a good
drawing of Gin the plane. We denote the maximum crossing number of Gas max -cr(G).
∗Louisiana State University, email: jfallo3@math.lsu.edu
†Colorado College, email: khogenson@coloradocollege.edu
‡Grand Valley State University, email: keoulaur@gvsu.edu
§Universidad Autonoma de San Luis Potos´ı, email: lomeli@ifisica.uaslp.mx
¶DePaul University, email: MSchaefer@cdm.depaul.edu
kBaruch College, CUNY email: pablo.soberon-bravo@baruch.cuny.edu
1

For the remainder of this paper, all drawings will be good drawings in the plane. By definition a graph
Gwith medges can have at most m
2crossings in a good drawing. Since incident edges cannot cross each
other in a good drawing, the actual number of crossing will be even smaller in general. This observation
gives us a natural upper bound to the maximum crossing number of a graph.
Definition. The thrackle bound for a graph Gwith edge set E(G) and vertex set V(G) is
ϑ(G) = |E(G)|
2−X
u∈V(G)deg(u)
2
The thrackle bound is an upper bound for the maximum crossing number [24], as it counts the number
of crossings we would have if every pair of non-incident edges crossed. A graph is called thrackleable if it has
a drawing which achieves the thrackle bound.
Among the most difficult problems concerning maximum crossing numbers is Conway’s thrackle conjec-
ture. Conway conjectures that a thrackleable graph on nvertices cannot have more than nedges. As of
2017, Conway is offering $1000 for a proof [10], which is up from his initial bounty of ten shillings and six
pence offered in 1969 [32]. Much work has been done on this conjecture [15, 22]. At the time of this note’s
writing, the most recent result posted on arXiv shows that any thrackleable graph on nvertices has at most
1.3984nedges [16]. Conway’s thrackle conjecture is true for the rectilinear drawings [32].
Another interesting open problem concerning maximum crossing numbers is the monotonicity conjecture.
For most variations of crossing numbers, monotonicity holds. That is, if His a subgraph of Gthen φ(H) is
at most φ(G) where φis the crossing number variant. The most striking exception is the maximum crossing
number, for which monotonicity is an open problem even for induced subgraphs [27]. Monotonicity is known
to hold for rectilinear drawings, a result due to Ringeisen, Stueckle, and Piazza [26].
Definition. Arectilinear drawing of a graph Gis a drawing in which all edges are straight line segments.
The maximum rectilinear crossing number of a graph G, max -cr(G), is the maximum number of crossings
in any rectilinear drawing of G.
The maximum rectilinear crossing number has been rediscovered several times under different names,
including obfuscation complexity [30].
Since max -cr(G)≤max -cr(G), the thrackle bound is also an upper bound for the maximum rectilinear
crossing number. Conway raised the problem of classifying all thrackleable graphs. In [32], Woodall gives a
complete classification assuming that the thrackle conjecture is true. He also considers the rectilinear case.
In Woodall’s classification of graphs with rectilinear drawings that obtain the thrackle bound, he finds the
exact maximum rectilinear crossing number for a particular spider. A spider is a subdivision of K1,k. Each
path radiating out from the degree kvertex of the spider is a leg. In general, the legs of the spider may be
of different lengths. For the spider with klegs in which each leg has length 2 we write S2
k. The graph S2
3is
typically called T2.
Theorem 1.1 (Woodall [32]).The graph S2
3does not have a rectilinear drawing that obtains the thrackle
bound. In fact, max -cr(S2
3) = 8 = ϑ(S2
3)−1.
This result contrats with the maximum crossing number case, since all trees are thrackleable. Our long-
term goal is to better understand the maximum rectilinear crossing numbers of trees, Theorem 1.1 shows
that this will require new ideas right from the start. In the current paper we focus on spider graphs, and use
Theorem 1.1 to compute an upper bound on the maximum rectilinear crossing number of spider graphs. In
Section 2 we provide an algorithm that gives a lower bound on the maximum rectilinear crossing number of
spiders with at least 3 legs. We conjecture that this algorithm gives the maximum rectilinear crossing number
for spiders. In Section 3 we use Mantel’s theorem to give an upper bound on max -cr(S) for any spider S.
Letting S2
kbe the spider in which each path is of length two, Woodall proves that max -cr(S2
3) = ϑ(S2
3)−1.
In Theorem 3.2 We extend this result by showing that max -cr(S2
k) = ϑ(S2
k)−k
2+jk2
4k.
For additional information regarding graph theory definitions and notation, we refer the reader to [31].
2

Previous Results
Finding the maximum rectilinear crossing number for a given graph is NP-hard [5], so it is not surprising
that there are only a few exact results for the maximum rectilinear crossing number of infinite families
of graphs; solved cases include complete k-partite graphs [20], cycles [28], and wheels [11], and there is a
lower bound for n-dimensional hypercubes which is conjectured to be exact [4]. The maximum rectilinear
crossing number has also been computed for C5×C5[25], the Petersen graph [12], and Q3, the 3-dimensional
hypercube [4]. For some other lesser known families of graphs, see [13, 14]. In another direction, some work
has been done to determine the maximum and minimum values of the maximum rectilinear crossing number
in a given family [7, 6, 8, 1, 2].
2 A Lower Bound
Let Sbe a spider with k≥3 legs L1, L2,...,Lk. Let ℓidenote the number of edges in leg Li, with
ℓ1≥ℓ2≥ · · · ≥ ℓk≥2. In this section we give an algorithm for constructing a rectilinear drawing of Sand
use it to find a lower bound on max -cr (S).
We begin by labeling the vertices of Sas follows. Label the degree-kvertex (0,0). Assign the label (i, j )
to the vertex on Liat distance jfrom (0,0). Next fix a point Pin the plane and place the vertices of S
anticlockwise on a circle centered at Paccording to the cyclic order [V0, V1, V2, V3, V4], where the vertices in
each subsequence Viare ordered as follows:
•V0: (0,0)
•V1: vertices with iodd and jeven, sorted by decreasing ithen increasing j
•V2: vertices with ieven and jodd, sorted by increasing ithen decreasing j
•V3: vertices with both iand jodd, sorted by decreasing ithen increasing j
•V4: vertices with both iand jeven and positive, sorted by increasing ithen decreasing j
After placing all vertices we draw the edges of Sas straight-line segments. If necessary, we perturb the
locations of the vertices to ensure our drawing is good.
Example 2.1. In Figure 1 we draw a spider with legs of length 4,3,2, and 2according to the algorithm.
Call this graph S. There are ϑ(S)−2 = 40 crossings. The two missed crossings are between the edges
(0,0) −(2,1) and (4,1) −(4,2) and between the edges (0,0) −(1,1) and (3,1) −(3,2). Every other pair of
non-incident edges crosses.
This drawing algorithm establishes the following lower bound on all spiders.
Proposition 2.2. Let Sbe a spider with k≥3legs of lengths ℓ1≥ℓ2≥ · · · ≥ ℓk. Then
max -cr (S)≥ϑ(S)−
k
X
i=3
(ℓi−1) i−1
2.
Proof. We claim that each pair of nonadjacent edges belonging to the same leg must cross each other. We
note that deleting V0and labeling (0,0) as (i, 0) to place it in V1or V4is consistent with the cyclic ordering
of vertices of Li. Let e1= (i, j1)-(i, j1+ 1) and e2= (i, j2)-(i, j2+ 1) be edges of Liwith j2> j1+ 1. We
note that both e1and e2have an end-vertex in each of Vmand Vm+2 for m= 1 or m= 2 (depending on the
parity of i) and e1’s end-vertices follow (or precede, again depending on the parity of i)e2’s in both Vmand
Vm+2 since jeither increases in both or decreases in both. In each case, e1and e2cross. For example, see
Figure 2.
Next we will consider the case where e1and e2are nonadjacent edges on different legs. Let e1= (i1, j1)-
(i1, j1+ 1) and e2= (i2, j2)-(i2, j2+ 1) with i16=i2. We distinguish three cases: (1) i1and i2(the leg
labels) have different parities, (2) i1and i2have the same parity with j1, j2both positive, and (3) i1and i2
have the same parity, but j1= 0.
3

(0,0) (1,1) (1,2) (1,3) (1,4)
(2,1)
(2,2)
(2,3)
(3,1)(3,2)
(4,1)
(4,2) V0
(0,0)
V1
(3,2)
(1,2)
(1,4)
V2
(2,3)
(2,1)
(4,1)
V3
(3,1)
(1,1)
(1,3)
V4
(2,2)
(4,2)
Figure 1: At left, a spider labeled as described in the algorithm. At right, the same spider drawn as described
in the algorithm.
Case 1. Suppose i1and i2have different parity. At most one of e1and e2may have an end vertex (0,0),
which we may consider without loss of generality to be the first vertex of V1or the last vertex of V4. In this
case, one of e1or e2has end vertices in V1and V3and the other has end vertices in V2and V4. Thus, e1and
e2cross because their end vertices alternate in cyclic order.
Case 2. Suppose i1and i2have the same parity and both j1and j2are positive. Without a loss of generality,
assume i1< i2. Then by construction, the cyclic order of the end vertices is (i2, odd)-(i1, even)-(i2, even)-
(i1, odd), so e1and e2cross.
Case 3. Suppose e1= (0,0)-(i1,1), i1and i2have the same parity, and i2is positive. We will consider two
subcases.
Case 3.1. If i1> i2, then there are two possibilities:
•e1’s end vertices are in V0and V2and e2’s are in V2(preceding (i1,1)) and V4.
•e1’s end vertices are in V0and V3and e2’s are in V3(following (i1,1)) and V1.
Either way, e1and e2cross.
Case 3.2. If i1< i2, then the end vertices of e1and e2do not alternate since iincreases in V2and decreases
in V3. This means that e1and e2do not cross.
The only time that crossings are missed is in Case 3.2. In particular, we have ℓi2−1 missed crossings
between Li1and Li2. As there are i2−1
2such pairs i1, i2, each i > 2 contributes (ℓi−1) i−1
2missed
crossings.
We have given a convex drawing, in which vertices are in convex position, of that gives a lower bound
on the maximum rectilinear crossing number of spiders. It was conjectured that every graph has a convex
rectilinear drawing maximizing the rectilinear crossing number [3]. While this has been disproven [9], it is
still open for trees.
4

P
V0
V1
V2
V3
V4(i, j1)
(i, j2+ 1)
(i, j1+ 1)
(i, j2)
Figure 2: Two edges from the same leg cross when iand j2are odd and j1is even.
3 An Upper Bound
Our lower bound on the maximum rectilinear crossing number of a spider was achieved by constructing
a good drawing of the spider. The logic here is reversed from what it is for the crossing number, where a
specific drawing would show an upper bound. This suggests that obtaining an upper bound on the maximum
crossing number (rather like the lower bound on the crossing number) is the tougher problem, since it requires
an argument applying to all good drawings of the graph. We work with an auxillary graph, in which non-
edges correspond to non-thrackled pairs of spider legs guaranteed by Theorem 1.1. For each non-edge of the
auxiliary graph we subtract one from the thrackle bound to give our upper bound. Our result is stated in
the following proposition.
Proposition 3.1. Let Sbe a spider with k≥3legs and assume each leg has length at least 2. Then
max -cr(S)≤ϑ(S)−k
2−k2
4.
Proof. Begin with a rectilinear drawing Dof S. Construct an auxiliary graph GS(D) on kvertices in which
each vertex corresponds to a particular leg of Sand each edge corresponds to a pair of legs which are
thrackled in D. A pair of legs is thrackled if any two nonadjacent edges in the two legs cross.
We claim GS(D) is triangle-free. If not, we would have three pairwise-thrackled legs of Sin D, which
implies that a subgraph of Sisomorphic to S2
3is thrackled in D. This contradicts Theorem 1.1.
Since GS(D) is triangle-free, Mantel’s Theorem [23] implies that GS(D) contains at most jk2
4kedges.
The minimum number of nonedges in GS(D) is thus k
2−jk2
4k. Since nonedges in GS(D) correspond to
non-thrackled leg pairs, and each non-thrackled leg pair is missing at least one crossing, we get that
max -cr(S)≤ϑ(S)−k
2−k2
4,
as desired.
We can now state an exact value for the maximum rectilinear crossing number of any spider in which all
legs have length two.
Theorem 3.2. Let S2
kbe a spider with k≥3legs where each leg has length 2. Then
max -cr(S2
k) = 2k
2−2k
2−k+k2
4.
5

Proof. First observe that S2
khas n= 2k+ 1 vertices and m= 2kedges. Additionally, one vertex of S2
khas
degree k,kvertices have degree 2, and kvertices have degree 1. Therefore, the thrackle bound for S2
kis
ϑ(S2
k) = 2k
2−k
2−k.
According to Proposition 3.1,
max -cr(S2
k)≤ϑ(S)−k
2−k2
4
=2k
2−2k
2−k+k2
4.
Further, according to Proposition 2.2,
max -cr(S2
k)≥ϑ(S2
k)−
k
X
i=3 i−1
2
=ϑ(S2
k)−(2P(k−2)/2
i=1 ifor keven
k−1
2+ 2 P(k−3)/2
i=1 ifor kodd
=ϑ(S2
k)−k
2+(k2
4for keven
k2−1
4for kodd
=ϑ(S2
k)−k
2+k2
4
=2k
2−2k
2−k+k2
4.
Together, these upper and lower bounds give us the desired equality.
4 Future Work
While the bounds given by our drawing algorithm and Mantel’s theorem agree when each leg of the spider
has length 2, this method does not seem to generalize. In our proof of the upper bound, we may only subtract
one crossing from the thrackle bound for every triple of legs, but in a spider with longer legs our algorithm
misses many more crossings. We believe that these crossings must be missed, and thus we conjecture that
the lower bound given in Proposition 2.2 is correct.
Conjecture 4.1. If Sis a spider with k≥3legs of lengths ℓ1≥ℓ2≥ · · · ≥ ℓk, then
max -cr(S) = ϑ(S)−
k
X
i=3
(ℓi−1) i−1
2.
Further, if our lower bound is correct, then we have exhibited a convex rectilinear drawing which achieves
the maximum rectilinear crossing number. This would prove a special case of the conjecture in [3]. Though
the conjecture is not true in general [9], we believe that there is a convex drawing obtaining the maximum
rectilinear crossing number for spiders and possibly trees in general.
A difficult problem that is of interest to us is finding the maximum rectilinear crossing number for trees.
To this end, it would be nice to find a proof for Conjecture 4.1. If that can be done, some additional tree
families to study are double spiders, which consist of two spider graphs with a single shared leg, and spiders
whose legs have been replaced by caterpillars.
6

As mentioned in the introduction, there is some research done on finding extremal values for the maximum
rectilinear crossing number in some graph families. It would be interesting to know which tree was the furthest
from thrackleable. That is, what is the minimum value among the maximum rectilinear crossing numbers of
trees on nvertices?
5 Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant Number DMS
1641020. Sober´on’s research is also supported by the National Science Foundation Grant Number DMS
1764237. We would like to thank the American Mathematical Society, ´
Eva Czabarka, Silvia Fern´andez-
Merchant, Gelasio Salazar, and L´aszl´o A. Sz´ekely for organizing the Mathematics Research Communities
workshop “Beyond Planarity”.
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