A conjecture of Richter and Salazar about graphs that are critical for a
fixed crossing number $k$ is that they have bounded bandwidth. A weaker
well-known conjecture of Richter is that their maximum degree is bounded in
terms of $k$. In this note we disprove these conjectures for every $k\ge 171$,
by providing examples of $k$-crossing-critical graphs with arbitrarily large
maximum degree

Bojan Mohar

DeVos

Geelen

Hliněný

Richter

Robertson

Zdeněk Dvořák

English

arXiv

AbstractA conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k⩾171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree

Dvořák, Zdeněk

Mohar, Bojan

Elsevier - Publisher Connector 

Crossing-critical graphs with large maximum degree 

Journal of Combinatorial Theory, Series B 100 (2010) 413–417Contents lists available at ScienceDirect
Journal of Combinatorial Theory,
Series B
www.elsevier.com/locate/jctb
Note
Crossing-critical graphs with large maximum degree
Zdeneˇk Dvorˇák 1,2, Bojan Mohar 3,4
Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6, Canada
a r t i c l e i n f o a b s t r a c t
Article history:
Received 16 March 2009
Available online 11 December 2009
Keywords:
Crossing number
Critical graph
Maximum degree
A conjecture of Richter and Salazar about graphs that are critical
for a ﬁxed crossing number k is that they have bounded bandwidth.
A weaker well-known conjecture of Richter is that their maximum
degree is bounded in terms of k. In this note we disprove these
conjectures for every k 171, by providing examples of k-crossing-
critical graphs with arbitrarily large maximum degree.
© 2009 Elsevier Inc. All rights reserved.
A graph is k-crossing-critical (or simply k-critical) if its crossing number is at least k, but every
proper subgraph has crossing number smaller than k. Using the Excluded Grid Theorem of Robertson
and Seymour [9], it is not hard to argue that k-crossing-critical graphs have bounded tree-width [2].
However, all known constructions of crossing-critical graphs suggested that their structure is “path-
like”. Salazar and Thomas conjectured (cf. [2]) that they have bounded path-width. This problem was
solved by Hlineˇný [3], who proved that the path-width of k-critical graphs is bounded above by 2 f (k) ,
where f (k) = (432 log2 k + 1488)k3 + 1.
In the late 1990s, two other conjectures were proposed and made public in 2003 at the Bled’03
conference [7] (see also [8,6]).
Conjecture 1. (See Richter [7].) For every positive integer k, there exists an integer D(k) such that every k-
crossing-critical graph has maximum degree less than D(k).
The second conjecture was proposed as an open problem in the 1990s by Carsten Thomassen and
formulated as a conjecture by Richter and Salazar.
E-mail addresses: rakdver@kam.mff.cuni.cz (Z. Dvorˇák), mohar@sfu.ca (B. Mohar).
1 Supported in part through a postdoctoral position at Simon Fraser University.
2 On leave from: Institute of Theoretical Informatics, Charles University, Prague, Czech Republic.
3 Supported in part by the Research Grant P1-0297 of ARRS (Slovenia), by an NSERC Discovery Grant (Canada) and by the
Canada Research Chair program.
4 On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.0095-8956/$ – see front matter © 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jctb.2009.11.003
414 Z. Dvorˇák, B. Mohar / Journal of Combinatorial Theory, Series B 100 (2010) 413–417Conjecture 2. (See Richter and Salazar [7,8].) For every positive integer k, there exists an integer B(k) such
that every k-crossing-critical graph has bandwidth at most B(k).
Conjecture 2 would be a strengthening of Hlineˇný’s theorem about bounded path-width and would
also imply Conjecture 1.
Hlineˇný and Salazar [5] recently made a step towards Conjecture 1 by proving that k-crossing-
critical graphs cannot contain a subdivision of K2,N with N = 30k2 + 200k.
In this note we give examples of k-crossing-critical graphs of arbitrarily large maximum degree,
thus disproving both Conjectures 1 and 2.
A special graph is a pair (G, T ), where G is a graph and T ⊆ E(G). The edges in the set T are called
thick edges of the special graph. A drawing of a special graph (G, T ) is a drawing of G such that the
edges in T are not crossed. The crossing number cr(G, T ) of a special graph is the minimum number
of edge crossings in a drawing of (G, T ) in the plane. (We set cr(G, T ) = ∞ if a thick edge is crossed
in every drawing of G .) An edge e ∈ E(G) \ T is k-critical if cr(G, T )  k and cr(G − e, T ) < k. Let
critk(G, T ) be the set of k-critical edges of (G, T ). If T = ∅, then we write just cr(G) for the crossing
number of G and critk(G) for the set of k-critical edges of G . Note that the graph G is k-critical if
critk(G) = E(G).
A standard result (see, e.g., [1]) is that we can eliminate the thick edges by replacing them with
suﬃciently dense subgraphs. (In fact, one can replace every edge xy by t = cr(G, T )+ 1 parallel edges
or by K2,t if multiple edges are not desired.)
Lemma 3. For every special graph (G, T ) with cr(G, T ) < ∞ and for any k, there exists a graph G˜ ⊇ G such
that cr(G, T ) = cr(G˜) and critk(G, T ) ⊆ critk(G˜).
Furthermore, note the following:
Lemma 4. Let k be an integer. Any graph G with cr(G) k contains a k-crossing-critical subgraph H such that
critk(G) ⊆ E(H).
Proof. For a contradiction, suppose that G is a smallest counterexample. If G were k-critical, then
we would set H = G , hence G contains a non-k-critical edge e. It follows that cr(G − e)  k. Let f
be a k-critical edge in G , i.e., cr(G − f ) < k. As cr((G − e) − f )  cr(G − f ) < k, f is a k-critical
edge in G − e. Therefore, critk(G) ⊆ critk(G − e). Since G is the smallest counterexample, G − e has
a k-critical subgraph H with critk(G − e) ⊆ E(H). However, H ⊆ G and critk(G) ⊆ E(H), which is a
contradiction. 
Let us now proceed with the main result. Two paths P1 and P2 in a special graph are almost
edge-disjoint if all the edges in E(P1) ∩ E(P2) are thick.
Lemma 5. For any d, there exist a special graph (G, T ) and a vertex v ∈ V (G) such that crit171(G, T ) contains
at least d edges incident with v.
Proof. Let (G, T ) be the special graph drawn as follows: we start with d+1 thick cycles C0,C1, . . . ,Cd
intersecting in a vertex v , i.e., Ci ∩ C j = {v} for 0  i < j  d. Their lengths are |C0| = 28, |Cd| = 24
and |Ci | = 7 for 1  i < d. They are drawn in the plane so that all their vertices are incident with
the unbounded face and their clockwise order around v is C0,C1, . . . ,Cd . See Fig. 1 illustrating the
case d = 5. Let C0 = va1a2 . . .a19b1b2b3c01c02 . . . c05, Cd = vtdb′3b′2b′1a′1a′2 . . .a′19 and Ci = vtici1ci2 . . . ci5 for
1 i < d. Furthermore, add d vertices s1, . . . , sd adjacent to v . The clockwise cyclic order of the neigh-
bors of v is a1, c05, s
1, t1, c15, s
2, t2, c25, . . . , s
d−1, td−1, cd−15 , sd, td,a′19. For 1 i  d, add thick cycles Ki
whose vertices in the clockwise order are ti , si , and ﬁve new vertices c˜i−15 , c˜
i−1
4 , . . . , c˜
i−1
1 . Finally, add
the following edges: cij c˜
i
j for 0  i < d and 1  j  5, aia′i for 1  i  19 and bib′i for 1  i  3. As
described, T =⋃di=0 E(Ci) ∪
⋃d
i=1 E(Ki). Let M = {a1a′1,a2a′2, . . . ,a19a′19,b1b′1,b2b′2,b3b′3}.
Z. Dvorˇák, B. Mohar / Journal of Combinatorial Theory, Series B 100 (2010) 413–417 415Fig. 1. A special graph with critical edges vsi .
This drawing G of (G, T ) has (192
) = 171 crossings, as the edges aia′i and a ja′j intersect for each
1  i < j  19, and there are no other crossings. Let us show that cr(G, T ) = 171. Let G′ be an
arbitrary drawing of (G, T ), and for a contradiction assume that it has less than 171 crossings. Let
us ﬁrst observe that every thick cycle Ci and K j is an induced nonseparating cycle of G . Therefore
it bounds a face of G′ . Consider the cyclic clockwise order of the neighbors of v according to the
drawing G′ . For each cycle Ci (0  i  d), the two edges of Ci incident with v are consecutive in
this order, since Ci bounds a face. Without loss of generality, we assume that each cycle Ci bounds
a face distinct from the unbounded one. If the cyclic order of the vertices around the face Ci is the
same as in the drawing G , we say that Ci is drawn clockwise, otherwise it is drawn anti-clockwise.
We may assume that C0 is drawn clockwise. If Cd were drawn clockwise as well, then each pair of
edges aia′i and a ja
′
j with 1  i < j  19 would intersect, and the drawing G′ would have at least
416 Z. Dvorˇák, B. Mohar / Journal of Combinatorial Theory, Series B 100 (2010) 413–417Fig. 2. A drawing of the graph G − vs3 with 170 intersections.
171 crossings. Therefore, Cd is drawn anti-clockwise. It follows that the edges aia′i and b jb
′
j intersect
for 1  i  19 and 1  j  3, and the edges bib′i and b jb′j intersect for 1  i < j  3, giving 60
crossings. For 1  i  5, let Pi be the path c0i c˜0i c˜0i−1 . . . c˜01t1c11c12 . . . c1i c˜1i . . . c˜11t2 . . . td . These paths are
mutually almost edge-disjoint and each of them intersects all edges of M in the drawing G′ , thus
contributing at least 110 crossings all together. Therefore, the drawing G′ has at least 170 crossings.
Since we assume that this drawing has fewer than 171 crossings, we conclude that there are no other
crossings.
The cycle va1a′1a′2 . . .a′19v splits the plane into two regions R1 and R2, such that R1 contains the
face bounded by C0 and R2 contains the face bounded by Cd . For j = 1,2, let A j be the set of cycles
Ci (0 i  d) such that the face bounded by Ci lies in the region R j . As P1 intersects the edge a1a′1
only once, A1 = {C0,C1, . . . ,Ck−1} and A2 = {Ck,Ck+1, . . . ,Cd} for some n with 1 n d. As the path
P1 does not intersect itself, all cycles in A1 are drawn clockwise and their clockwise order around
v is C0,C1, . . . ,Cn−1. Similarly, all cycles in A2 are drawn anti-clockwise and their clockwise order
around v is Cd,Cd−1, . . . ,Cn .
Let us now consider the cycle Kn . Since the edges c
n−1
4 c˜
n−1
4 and c
n−1
5 c˜
n−1
5 do not intersect, the
thick path cn−15 vtksnc˜
n−1
5 is not intersected, and Cn−1 is drawn clockwise, Kn is drawn clockwise
as well. Since Cn lies in the region R2, the vertex tn and thus the whole thick cycle Kn lie in R2.
However, that means that the edge skv intersects either the path P1 or the edge a1a′1, which is a
contradiction. We conclude that cr(G, T ) = 171.
Z. Dvorˇák, B. Mohar / Journal of Combinatorial Theory, Series B 100 (2010) 413–417 417On the other hand, cr(G − vsn, T ) < 171, for 1  n  d (in fact, cr(G − vsn, T ) = 170). To see
that, consider the drawing of (G − vsn, T ) in which the cycles C0,C1, . . . ,Cn−1 are drawn clockwise,
the cycles Cn,Cn+1, . . . ,Cd are drawn anti-clockwise, and the cyclic order of the neighbors of v is
a1c05s
1t1c15 . . . s
n−1tn−1cn−15 a′19tdc
d−1
5 s
d−1td−1 . . . cn5tn , see Fig. 2. The intersections of this drawing are
of edges aia′i with b jb
′
j for 1  i  19 and 1  j  3, the edges bib′i with b jb′j for 1  i < j  3,
and the edges cn−1i c˜
n−1
i with all edges of M for 1 i  5. Therefore, the edge vsn is 171-critical for
each n, so v is incident with d critical edges. 
We are ready for our main result.
Theorem 6. For every k  171 and every d, there exists a k-crossing-critical graph H containing a vertex of
degree at least d.
Proof. Let (G, T ) be the special graph constructed in Lemma 5. By Lemma 3, there exists a graph
H ′ ⊇ G such that cr(H ′) = cr(G, T )  171 and crit171(G, T ) ⊆ crit171(H ′). Let H be the 171-critical
subgraph of H ′ obtained by Lemma 4. As crit171(G, T ) ⊆ crit171(H ′) ⊆ E(H), H contains at least d
edges incident with one vertex, hence (H) d. For k > 171 we add to H k − 171 disjoint copies of
K5 in order to get a k-crossing-critical graph. 
Actually, in the proof of Theorem 6, we can take t = 
 k171  copies of the graph H and k − 171t
copies of K5. This gives rise to a k-critical graph with t = Ω(k) vertices of (arbitrarily) large degree.
We conjecture that this is best possible in the following sense:
Conjecture 7. For every positive integer k there exists an integer D = D(k) such that every k-crossing-critical
graph contains at most k vertices whose degree is larger than D.
It is not even obvious that there exist k-crossing-critical graphs with arbitrarily many vertices of
degree more than 6. Surprisingly, such examples have been constructed recently by Hlineˇný [4]. His
examples may contain arbitrarily many vertices of any even degree smaller than 2k − 1.
Let us also remark that the use of Lemma 4 means that we do not present an explicit counterexam-
ple to Conjecture 1, but only its supergraph. Consequently, we cannot ensure that the counterexample
has some particular properties, e.g., we cannot prove that Conjecture 1 fails for 3-connected sim-
ple graphs. It might be of interest to rectify this problem by carrying out the construction explicitly,
replacing the cycles of thick edges by some suitable planar graphs.
References
[1] M. DeVos, B. Mohar, R. Šámal, Unexpected behaviour of crossing sequences, Electron. Notes Discrete Math. 31 (2008) 259–
264.
[2] J.F. Geelen, R.B. Richter, G. Salazar, Embedding grids in surfaces, European J. Combin. 25 (2004) 785–792.
[3] P. Hlineˇný, Crossing-number critical graphs have bounded path-width, J. Combin. Theory Ser. B 88 (2003) 347–367.
[4] P. Hlineˇný, New inﬁnite families of almost-planar crossing-critical graphs, Electron. J. Combin. 15 (2008) #R102.
[5] P. Hlineˇný, G. Salazar, Stars and bonds in crossing-critical graphs, preprint, 2008.
[6] B. Mohar, J. Pach, B. Richter, R. Thomas, C. Thomassen, Topological graph theory and crossing numbers, Report on the BIRS
5-Day Workshop, 2007, 18 pages, http://www.birs.ca/workshops/2006/06w5067/report06w5067.pdf.
[7] R.B. Richter, Problem 437, in: Research Problems from the 5th Slovenian Conference, Bled, 2003, Discrete Math. 207 (2007)
650–658.
[8] R.B. Richter, G. Salazar, A survey of good crossing number theorems and questions, in press.
[9] N. Robertson, P.D. Seymour, Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986) 92–114.


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