Ryo Nikkuni Regular Projections of Spatial Graphs
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Regular Projections of Spatial Graphs
Ryo NIKKUNI
ABSTRACT
Study of the regular projection in spatial graph theory is more rich in content as compared withthe regular projection in knot theory. Actually some interesting phenomena in the regular projectionof spatial graphs which are not appeared in the one of knots and links have been discovered Since1990s. In this article we introduce the recent topics in the regular projection of spatial graphs,in particular, knotted projection (KP), identifiable projection (IP) and completely distinguishableprojection (CDP). We also present some unsolved problems in this area.
1. Introduction
1.1. KP, IP and CDP
Let G be a finite graph. We give a label to each of vertices and edges of G and denote
the set of all vertices and the set of all edges of a graph G by V(G) and E(G), respectively.
An embedding of G into R3 is called a spatial embedding of G or simply a spatial graph.
Two spatial embeddings f and g of G are said to be ambient isotopic if there exists an
orientation preserving homeomorphism : R3 R3 such that f = g. A graph G
is said to be planar if there exists an embedding of G into R2, and a spatial embedding
of a planar graph G is said to be trivial if it is ambient isotopic to an embedding of G
into R2 R3. We note that a trivial spatial embedding of a planar graph is unique up
to ambient isotopy in R3 [14].
A regular projectionof G is an immersion from G to R2 whose multiple points are only
finitely many transversal double points away from vertices. For a regular projection f of
G with p double points, we can obtain 2p regular diagrams of the spatial embeddings of
G from f by giving over/under information to each of the double points. Then we say
that a spatial embedding f of G is obtained from f if f is represented by one of these 2p
regular diagrams. At this time we also call f a regular projection of f.
Let f be a regular projection of a graph which is homeomorphic to the disjoint union
of 1spheres. Then it is well known that a trivial spatial embedding, namely a trivial
link, can be always obtained from f. This fact plays an important role in knot theory,
e.g. the theory of skein polynomial invariants is based on this fact. But this fact does
not always hold for a regular projection of a planar graph. For example, let G be the
octahedron graph and f a regular projection of G as illustrated in Fig. 1.1.1. Taniyama
pointed out that any of the spatial embeddings of G obtained from f is nontrivial [27].
Such a regular projection is said to be knotted, namely a regular projection
f of a planar
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graph G is said to be knotted [27] if any of the spatial embeddings of G obtained from
f is nontrivial. We call a knotted regular projection a KP simply. The existence of a
KP is one of the suggestive phenomena in spatial graph theory. In the next section, we
introduce some fundamental results and the recent development in the study of KP.
f
3
G
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2
1
6
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1
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62
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Fig. 1.1.1. A knotted regular projection
While a KP detects a nontriviality of a regular projection, let us consider a unifor
mity of a regular projection. A regular projection f of G is said to be identifiable [10] if
all regular diagrams of the spatial embeddings of G obtained from f represent mutually
ambient isotopic spatial embeddings of G. For example, each of the regular projections
as illustrated in Fig. 1.1.2 (1), (2) and (3) is identifiable. Actually we can see that only
trivial spatial embeddings can be obtained from each of these regular projections. We
call an identifiable regular projection an IP simply. As the examples above suggest, an IPdetects a triviality of a regular projection. Namely, an IP cannot be a KP. We explain
the details in section 3.
(1) (2) (3)
1 2
1
4 3
2
Fig. 1.1.2. Identifiable regular projections
On the other hand, a regular projection f of G is said to be completely distinguishable
[17] if all regular diagrams of the spatial embeddings of G obtained from f represent
mutually different spatial embeddings of G up to ambient isotopy. For example, let us
recall a regular projection f of the octahedron graph as illustrated in Fig. 1.1.1. As we
said before,f
is a KP, namely we can obtain the eight spatial embeddings ofG
from
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f as illustrated in Fig. 1.1.3 and it can be checked that they are nontrivial spatial
embeddings. Moreover, by observing the constituent knots and links carefully, we can
see that these eight spatial embeddings are mutually not ambient isotopic. Thus f is
completely distinguishable. We call a completely distinguishable regular projection aCDP simply. As the example above suggests, a CDP of a planar graph is related to
KP. Moreover, a CDP of a nonplanar graph can be also considered and relates to the
minimal crossing number of a graph. In section 4, we introduce the recent results in the
study of CDP.
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1
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Fig. 1.1.3.
2. Knotted projection (KP)
2.1. Trivializable graphs
A planar graph is said to be trivializable if the graph does not have a KP. For example, a
graph homeomorphic to the disjoint union of 1spheres is trivializable, and the octahedron
graph is not trivializable.
Question 2.1.1. When is a planar graph trivializable?
It is known that there exist infinitely many trivializable graphs. Taniyama showed that
every bifocal as illustrated on the lefthand side in Fig. 2.1.1 is trivializable [27]. On
the other hand, Sugiura and Suzuki showed that every 3line web as illustrated on the
righthand side in Fig. 2.1.1 is trivializable [23]. Tamura showed that every neobifocalis
trivializable and gave a systematic construction of trivializable graphs in terms of an edge
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sum of graphs [24].1 We refer the reader to [27], [24] and [23] for the precise definitions
of the bifocal, the neobifocal and the 3line web, respectively. But trivializable graphs
have not been characterized completely yet.
i
Fig. 2.1.1. A bifocal and a 3line web
2.2. Forbidden graphs for the trivializability
We investigate trivializable graphs from a stand point of graph minor theory. A graph
H is called a minor of a graph G if H can be obtained from G by a finite sequence of an
edge contraction or taking a subgraph. We say that a property P of a graph is inherited
by minors if a graph has P then each proper minor of the graph also has P. Let (P) be
the set of all graphs which do not have P and whose all proper minors have P. This set
is called the obstruction set for P and each of the elements in (P) is called a forbidden
graph for P. It is clear that a graph G has P if and only if G does not have a minor
which belongs to (P). Then, according to RobertsonSeymours Graph Minor Theorem
[20], the following fact holds.
Theorem 2.2.1. (RobertsonSeymour [20]) (P) is a finite set.
In particular for the trivializability T, it is known the following.
Proposition 2.2.2. (Taniyama [27]) The trivializability is inherited by minors.
Therefore by Theorem 2.2.1, we have that (T) is a finite set, namely the trivializability
of a graph can be determined by finitely many forbidden graphs. Thus we want to
determine all elements in (T), namely we consider the following problem.
Problem 2.2.3. Find all forbidden graphs for the trivializability.1 Her method can produce a trivializable graph which is not a minor of a bifocal or neobifocal. But the author does
not know whether her method can produce a trivializable graph which is not a minor of a 3line web or not.
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By the works of SugiuraSuzuki and the authorOzawaTaniyamaTsutsumi, sixteen
forbidden graphs for the trivializability have been found as follows.
Theorem 2.2.4. (1) (SugiuraSuzuki [23]) The seven graphs G1, G2, . . . , G7 as illus
trated in Fig. 2.2.1 belong to (T).
(2) (NikkuniOzawaTaniyamaTsutsumi [18]) The nine graphs G8, G9, . . . , G16 as illus
trated in Fig. 2.2.1 belong to (T).
G2 G3G1
G5 G6G4
G7 G9 G10G8
G12 G13G11
G15 G16G14
Fig. 2.2.1. Forbidden graphs G1, G2, . . . , G16 for the trivializability
Indeed, each Gi has a KP fi (i = 1, 2, . . . , 16) as illustrated in Fig. 2.2.2. Actually any
of the spatial embeddings of Gi obtained from fi contains a Hopf link (i = 1, 2, . . . , 16).
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Therefore each Gi is not trivializable. Moreover it can be seen that each of the proper
minors of Gi is also a minor of a 3line web. Thus by Proposition 2.2.2 we have that
G1, G2, . . . , G16 (T).
Now, we show a new forbidden graph for the trivializability. Let G17 be a planar graphand f17 a regular projection of G17 as illustrated in Fig. 2.2.3. We can see easily that
f17 is a KP, actually we can check that each of the spatial embeddings of G17 obtained
from f17 also contains a Hopf link. Thus G17 is nontrivializable. Moreover we have the
following.
Theorem 2.2.5. The graph G17 belongs to (T).
The proof is the same as the proof for G1, G2, . . . , G16, actually we can see that each
of the proper minors of G17 is a minor of a 3line web and G17 is not homeomorphic to
G1, G2, . . . , G16. So we omit the details.
It seems that (T) is not determined by these seventeen forbidden graphs. Actually
we have candidates for forbidden graphs for the trivializability. Let H be the graph as
illustrated in Fig. 2.2.4. Sugiura asked the following question in his master thesis [22].
Question 2.2.6. (Sugiura [22]) Is the graph H trivializable?
This question is still open. Since we can see that each of the proper minors of H is
trivializable, ifH is not trivializable then H (T). We remark here that for any regular
projection f of H there exists a spatial embedding f of H obtained from f such that f
does not contain any nontrivial link [18]. Therefore if H is nontrivializable then we can
find a spatial embedding f of H which is obtained from its KP and does not contain any
nontrivial link.
Besides, let H1, H2, H3 and H4 be four graphs as illustrated in Fig. 2.2.5. Then we
have that each Hi has a KP gi (i = 1, 2, 3, 4) as illustrated in Fig. 2.2.6.
Question 2.2.7. (NikkuniOzawaTaniyamaTsutsumi [18]) Is each of Hi a forbidden
graph for the trivializability?
Note that each of Hi has a minor which is homeomorphic to H. Thus if H is not
trivializable then H1, H2, H3 and H4 are not forbidden graphs for the trivializability, and
if H is trivializable then there is a possibility that H1, H2, H3, H4 (T).
2.3. Spatial subgraph in a spatial graph obtained from a KP
For the present, any spatial graph obtained from a known KP contains a Hopf link.
Then the following question can be considered in general.
Question 2.3.1. What kind of spatial subgraph does appear in a spatial graph obtained
from a KP?
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8f 9f 10f
11f 12f 13f
14f 15f 16f
7f
4f 5f 6f
1f 2f 3f
Fig. 2.2.2. A KP fi : Gi R2 (i = 1, 2, . . . , 17)
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f
G17
1
2
3
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6
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8 75
1
4 217
Fig. 2.2.3. New forbidden graph and its KP
H
Fig. 2.2.4.
H2 H3H1 H4
Fig. 2.2.5.
1g 2g 3g 4g
Fig. 2.2.6. A KP gi : Hi R2 (i = 1, 2, 3, 4)
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As a spacial case of Question 2.3.1, the following question was asked in [18].
Question 2.3.2. (NikkuniOzawaTaniyamaTsutsumi [18]) Let f be a KP of a planar
graph G and f a spatial embedding of G obtained from f. Does there exist a proper
subgraph H of G such that fH is a nontrivial spatial embedding?
We remark here that Question 2.3.2 is equivalent to the following question.
Question 2.3.3. (NikkuniOzawaTaniyamaTsutsumi [18]) Is a regular projection of a
minimally knotted spatial embedding of a planar graph not a KP?
Here a spatial embedding f of a planar graph G is said to be minimally knotted if f is
nontrivial and fH is trivial for any proper subgraph H of G. For example, each of the
spatial graphs f, g and h as illustrated in Fig. 2.3.1 is known as a minimally knotted
spatial graph.
In the following we give a partial answer for Question 2.3.3. A spatial embedding f
of a planar graph G is said to be strongly almost trivial [7] if f is nontrivial and there
exists a regular projection f of f such that any of the spatial embeddings of H obtained
from fH is trivial for any proper subgraph H ofG. At this time we also call f a strongly
almost trivial projection of G. It is clear that if f is strongly almost trivial then it is
minimally knotted. For example, each of the spatial graphs f and g as illustrated in
Fig. 2.3.1 is strongly almost trivial, actually they have f and g as strongly almost trivial
projections, respectively. But a spatial graph h as illustrated in Fig. 2.3.1 is not strongly
almost trivial [9]. Then we have the following.
Proposition 2.3.4. A strongly almost trivial projection of a planar graph is not a KP.
We prepare some definitions and lemmas that are needed to prove Proposition 2.3.4 (We
also need them in the next section). A spatial graph is said to be free if the fundamental
group of the spatial graph complement is a free group. Then the following is Scharlemann
Thompsons famous criterion.
Lemma 2.3.5. (ScharlemannThompson [21]) A spatial embedding of a planar graph is
trivial if and only if any of the spatial subgraphs is free.
On the other hand, the following is known.
Lemma 2.3.6. (NikkuniOzawaTaniyamaTsutsumi [18]) Let f be a regular projection
of a graph (does not need to be planar). Then there exists a free spatial embedding of the
graph obtained from f.
Now we give a proof of Proposition 2.3.4. The proof is essentially the same as the proof
of [18, Proposition 3.4]. Let f be a strongly almost trivial spatial embedding of a planar
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f g h
f g h
Fig. 2.3.1.
graph G and f the strongly almost trivial projection. Since f is minimally knotted, by
Lemma 2.3.5 we have that f is not free. On the other hand, by Lemma 2.3.6, there exists
a free spatial embedding g of G obtained from f. Then, by the definition of the strongly
almost triviality, we have that gH is trivial for any proper subgraph H of G, namely gHis free for any subgraph H of G. Thus by Lemma 2.3.5 we have that g is trivial. So f is
not a KP.
3. Identifiable projection (IP)
3.1. IP and the Simon invariant
When we consider an IP, the following fact is fundamental.
Proposition 3.1.1. (HuhTaniyama [10]) A nonplanar graph does not have an IP.
To explain Proposition 3.1.1, we recall the Wu invariant[30] of a spatial graph. Let X
be a topological space and
C2(X) = {(x, y) X X  x = y}
the configuration space of ordered two points of X. Let be an involution on C2(X)
defined by (x, y) = (y, x). Then we call the integral cohomology group of Ker(1 + #)
the skewsymmetric integral cohomology group of the pair (C2(X), ) and denote it by
H
(C2(X), ). It is known that H2(C2(R3), ) = Z [30]. We denote a generator of
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H2(C2(R3), ) by . Let f be a spatial embedding of a graph G. Then f induces a
homomorphism
(f2) : H2(C2(R3), ) H2(C2(G), ).
We call (f2)() the Wu invariant of f and denote it by L(f). In particular, if G is
homeomorphic to K5 or K3,3 then H2(C2(G), ) = Z [28] and L(f) coincides with the
Simon invariant [26]. We note that L(f) coincides with twice the linking number if G is
homeomorphic to the disjoint union of two 1spheres. We refer the reader to [28] for a
diagramatic calculation ofL(f).
The generator of H2(C2(R3), ) depends on the orientation of R3 [30], namely it
holds that L(f!) = L(f) for a spatial embedding f of G, where f! denotes the mirror
image off. Since the Simon invariant is odd integervalued [26], we have that any spatial
embedding of K5 and K3,3 is not ambient isotopic to its mirror image. This implies thatany spatial embedding of a nonplanar graph G is not ambient isotopic to its mirror
image because G always contains a subgraph which is homeomorphic to K5 or K3,3 [13].
Therefore a nonplanar graph cannot have an IP.
3.2. Characterizing spatial graphs obtained from an IP
Now let us consider an IP f of a planar graph G. If G is trivialzable, then a trivial
spatial embedding of G can be always obtained from f. Hence the following is clear by
the definition of the identifiability and the uniqueness of a trivial spatial embedding.
Proposition 3.2.1. Let G be a trivializable planar graph and f a regular projection
of G. Then, f is an IP if and only if any of the spatial embeddings obtained from the
projection is trivial.
By the existence of a KP, the proof of Proposition 3.2.1 does not work for non
trivializable graphs as it is. But by an application of Lemmas 2.3.5 and 2.3.6, we can
generalize Proposition 3.2.1 to arbitrary planar graphs as follows.
Theorem 3.2.2. (Nikkuni [16]) A regular projection of a planar graph is an IP if andonly if any of the spatial embeddings obtained from the projection is trivial.
Actually, we have the if part by the uniqueness of the trivial spatial embeddings of
a planar graph. And we have the only if part as follows. Let f be an IP of a planar
graph G and f the spatial embedding of G obtained from f. We note that fH is also
an IP for any subgraph H of G. Then by Lemma 2.3.6 the spatial embedding g of H
obtained from fH is free. Since g = fH, we have that fH is free for any subgraph H
of G. Therefore by Lemma 2.3.5 we have that f is trivial. This completes the proof of
Theorem 3.2.2.
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A regular projection of a graph is said to be reducedif the image of the projection has
no local parts as illustrated in Fig. 3.2.1 (a) and (b). For example, each of the regular
projections as illustrated in Fig. 1.1.2 (2) and (3) is reduced. Note that there exist
infinitely many planar graphs whose reduced IP are only embeddings from the graph toR
2, see [10] (We also refer the reader to [3], [19], [25], [11], [31] and [8] for related works).
Theorem 3.2.2 indicates that the spatial embedding of a planar graph obtained from an
IP of the graph is always trivial even if the projection is reduced and not an embedding
into R2.
(a) (b)
Fig. 3.2.1.
4. Completely distinguishable projection (CDP)
4.1. CDP of a planar graph
We recall that the knottedness of a regular projection is defined for only planar graphs,
and the identifiability of a regular projection is considered for only planar graphs as
a consequence of Proposition 3.1.1. But the complete distinguishability of a regular
projection can be considered not only for planar graphs but also for nonplanar graphs.
First we deal with the case of planar graphs in this subsection. A CDP of a graph is said
to be trivial if it has no double points. Therefore every planar graph has a trivial CDP.
Moreover, a nontrivial CDP of a planar graph is closely related to KP as follows.
Proposition 4.1.1. (1) LetG be a trivializable planar graph. Then a regular projection
f of G is a CDP if and only if it has no double points.(2) LetG be a nontrivializable planar graph. If a regular projectionf ofG is a nontrivial
CDP then f is a KP.
Actually it is clear that any spatial embedding of a graph which can be obtained from a
nontrivial completely distinguishable projection of the graph is not ambient isotopic to
its mirror image. By this fact, we can prove Proposition 4.1.1. We note that the converse
of Proposition 4.1.1 (2) is not true, see Examples 4.1.2 and 4.1.3.
Example 4.1.2. Let G be a nontrivializable planar graph and f a CDP of G. Note
that f is also a KP by Proposition 4.1.1 (2). Then by producing the local parts in f(G)
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as illustrated in Fig. 4.1.1, we can construct another KP which is not a CDP.
Fig. 4.1.1.
Example 4.1.3. Let G be a planar graph and f a regular projection ofG as illustrated
in Fig. 4.1.2. Since f(G) contains a image of a KP of the octahedron graph as illustrated
in Fig 1.1.1, we have that f is a KP. We can also check that f(G) does not contain any
local parts as in Fig. 4.1.1. But Fig. 4.1.3 shows that
f is not a CDP.
f
G
2 3
4
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9
7 8
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7 8
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1 1
2 3
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12 10
Fig. 4.1.2.
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12 10
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12 10
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12 10
Fig. 4.1.3.
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4.2. CDP of a nonplanar graph
In this subsection we deal with the case of nonplanar graphs. Let Kn be the complete
graph on n vertices and Km,n the complete bipartite graph on m + n vertices. Note that
Kn is nonplanar if n 5, and Km,n is nonplanar if min{m, n} 3.
Theorem 4.2.1. (Nikkuni [17]) Each of Kn and Km,n has a CDP.
We note that each of Kn for n 4 and Km,n for min{m, n} 2 is planar and trivializ
able [23]. So in this case only embeddings from the graph intoR2 are CDP by Proposition
4.1.1 (1).
In the following we show that Kn has a CDP. Since Kn has a trivial CDP for n 4, we
assume that n 5. We set V(Kn) = {1, 2, . . . , n}. We denote the edge ofKn connecting
two vertices i and j by ei,j (1 i < j n). Let Wn = Cn1 st(n) be a subgraph of Knembedded in R2 as illustrated in Fig. 4.2.1, where Cn1 = e1,2 e2,3 e1,n1 and
st(n) = n1i=1 ei,n. Then Cn1 divides R2 into two domains and we can construct a regular
projection f : Kn R2 by embedding each of the edges in E(Kn) E(Wn) into the non
compact domain such that any pair of the edges ei,j and ek,l (1 i < k < j < l n 1)
has exactly one double point and the other pair of the edges has no double points, see Fig.
4.2.1. Then, for the double point P between ei,j and ek,l (1 i < k < j < l n 1),
we can find a subgraph H = Cn1 ei,n ek,n ej,n el,n ei,j ek,l of Kn which is
homeomorphic to K5 such that fH is a regular projection ofH with exactly one double
point P. Let g and g! be exactly two spatial embeddings of H which can be obtainedfrom fH. Then we have that L(g) = L(g!) = 1. Since L(g!) = L(g), we have that
L(g) = L(g!). This implies that f is a CDP.
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3
45
7
61
2 3
4
5
1
2
3 4
5
6
Fig. 4.2.1. A CDP f : Kn R2 (n = 5, 6, 7)
We remark here that the construction above is based on a complete calculation of the
Wu invariants of spatial embeddings of Kn by the author, see [15] for the details.
Conjecture 4.2.2. Every graph has a CDP.
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Since a planar graph always has a trivial CDP, our aim is to prove whether a non
planar graph always has a CDP. For a planar graph, it is also natural to ask the following
question.
Question 4.2.3. Does a nontrivializable planar graph have a nontrivial CDP?
4.3. CDP and the minimal crossing projection
Let cr(f) be the number of double points of a regular projection f : G R2. Then we
call cr(G) = min{cr(f)  f : G R2 is a regular projection} a minimal crossing number
ofG. A regular projection f ofG is called a minimal crossing projectionifcr(f) = cr(G).
For example, cr(K5) = cr(K3,3) = 1, cr(K3,4) = 2 and cr(K6) = 3 (cf. [6]). Fig. 4.3.1
shows an example of the minimal crossing projections of K5, K3,3, K3,4 and K6.
Fig. 4.3.1. Minimal crossing projections
Let f be a CDP ofKn (resp. Km,n) constructed by the method in the proof of Theorem
4.2.1. Then it can be easily seen that f is not a minimal crossing projection if n 6
(resp. min{m, n} 3 and max{m, n} 4). On the other hand, we can check that each
of the minimal crossing projections as illustrated in Fig. 4.3.1 is a CDP in a similar
way as the proof of Theorem 4.2.1. From this viewpoint, Ozawa suggested the following
question.
Question 4.3.1. ([17]) Is a minimal crossing projection a CDP?
Note that a minimal crossing projection of a planar graph is an embedding of the graphinto R2, namely a trivial CDP. Besides we can give an affirmative answer to Question
4.3.1 for the case of minimal crossing number one [17].
We present a piece of circumstantial evidence as follows.
Example 4.3.2. Let f be the regular projection of K2n, which is BlazekKomans
construction of a candidate for the minimal crossing projection of K2n [1]. Fig. 4.3.2
illustrates the case of n = 4 and in this case f is a minimal crossing projection of K8.2
2 R. K. Guy conjectured in [4] that
cr(Kn) =1
4
n
2
n 1
2
n 2
2
n 3
2
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Then we can see that f is a CDP in a similar way as the proof of Theorem 4.2.1.
1
2 3
4
5
67
8
1
2 3
4
5
6 7
8
Fig. 4.3.2. f : K2n R2 (n = 4)
Example 4.3.3. Let f be the regular projection of Km,n, which is Zarankiewiczs con
struction of a candidate for the minimal crossing projection of Km,n [32]. Fig. 4.3.3
illustrates the case of m = n = 6 and in this case f is a minimal crossing projection of
K6,6.3 Then we can also see that f is a CDP in a similar way as the proof of Theorem
4.2.1.
1
3
5
7
9
11
2 4 6 8 10 12
1
3
5
7
9
11
2 4 6 8 10 12
Fig. 4.3.3. f : Km,n R2 (m = n = 6)
Conjecture 4.3.4. A minimal crossing projection is a CDP.
We remark here that if Conjecture 4.3.4 is solved affirmatively then Conjecture 4.2.2
is also solved affirmatively.
and showed that the conjecture above is true for n 10 [5].3 From the beginning, the crossing number problem for Km,n is called Tur ans brick factory problem (cf. [29]). In [32],
K. Zarankiewicz claimed that
cr(Km,n) = m
2
m 1
2
n
2
n 1
2
.
However Guy pointed out a gap for Zarankiewiczs proof [4], and now this is called Zarankiewiczs conjecture. At present,
Zarankiewiczs conjecture is true for m 6 [12].
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Department of Mathematics, Faculty of Education, Kanazawa University
Kakumamachi, Kanazawa, 9201192, Japan
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