Circular Chromatic Number of Even-Faced Projective Plane
Graphs
DRAFT 2: NOT FOR DISTRIBUTION
Luis Goddyn*
Department of Mathematics
Simon Fraser University
Burnaby, BC, Canada
E-mail: [email protected]
We derive an exact formula for the circular chromatic number of any graph
embeddable on the projective plane in such a way that all of it faces have even
length.
1. INTRODUCTION
The circular chromatic number χc (G) of a graph G, is an invariant which
refines the chromatic number χ(G). More precisely, if G is finite and loopless, then χc (G) is a rational number in [2, ∞) satisfying χ(G) = dχc (G)e.
Of the numerous equivalent definitions of χc (G) [37], perhaps the easiest
to state is the following. For any loopless graph G, χc (G) is the least real
number c such that there exists a function from V (G) to the set of unitlength open arcs of a circle of circumference c, such that adjacent vertices
map to disjoint arcs. If G has a loop, then we define χc (G) = ∞. It is
NP-hard to calculate χc (G), even if G is planar and χ(G) is part of the
input [?]. In this paper we give an exact, and quickly computable, formula
for χc (G) which applies to a particular class of graphs.
An even-faced projective plane graph is a graph which has been embedded in the projective plane in such a way that a walk around the boundary
of any face has even length. A surface subgraph of an embedded graph
G is any embedded graph which may be obtained from G by successively
deleting edges which bound distinct faces. We make these definitions pre-
* Supported in part by the National Sciences and Engineering Research Council of
Canada, and the Pacific Institute for the Mathematical Sciences
1
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cise in Section 2. Let maxfl(G) denote the maximum length of a face of an
embedded graph G. We may now state our main result.
Theorem 1.1. Let G be an even-faced projective plane graph. Then
either G is bipartite (in which case χc (G) = 2), or
χc (G) = 2 +
2
r−1
where 2r is the least value of maxfl(H) among all surface subgraphs H of G.
In particular, every even-faced projective plane graph has circular chromatic number 2 + 2/s for some s ∈ {0, 1, . . . , ∞}. (We interpret 2/0 = ∞
and 2/∞ = 0.)
The case k = 1 of Theorem 1.1 corresponds to G having a loop, which
is necessarily noncontractible. That loop induces a surface subgraph H for
which maxfl(H) = 2.
The case k = 2 strengthens a result of Youngs [35]. He shows that
loopless quadrangulations G of the projective plane satisfy χc (G) ∈ {2} ∪
(3, 4] whereas our result implies that χc (G) ∈ {2, 4}. An application of this
regards the well-known Mycielski transformation [23]: given a graph G,
we join a new copy v 0 of each v ∈ V (G) to the neighbours of v, then
we join a new vertex x to all vertices of the form v 0 . This constructs,
from any simple graph G, a new graph denoted µ(G) having the same
clique number as G, and such that χ(µ(G)) = χ(G) + 1. For any odd
circuit C2k+1 , the graph µ(C2k+1 ) embeds on the projective plane as a
nonbipartite quadrangulation. (Here µ(C5 ) is the famous Grötzsch graph.)
Our results imply that χc (µ(C2k+1 )) = 4 even though χc (C2k+1 ) = 2+1/k.
This result already appears in [4]. We add that a similar statement holds
true for the ‘generalized Mycielski construction’, a variation which which
has recently gained attention with regard to graph homomorphisms [29].
When applied to an odd circuit, the generalized Mycielski construction also
results in a nonbipartite quadrangulation whose circular chromatic number
is exactly 4.
The case k = 3 implies that any loopless nonbipartite even-faced projective plane graph G containing no quadrangulation has circular chromatic
number 3. Every hexangulation of the projective plane satisfies χc (G) ∈
{2, 3, 4, ∞}, whereas octangulations satisfy χc (G) ∈ {2, 8/3, 3, 4, ∞}.
A related result of Gimbel and Thomassen [10] states that all projective
plane graphs with girth at least 5 satisfy χc (G) ≤ 3. Devos et. al. [6]
consider even-faced graphs G embedded on an arbitrary surface. They
show that, provided G has large edge-width, χc (G) is either very close to
2, or χc (G) ≥ 2 + 2/(k − 1) where 2k = maxfl(G). This paper shows that
their lower bound is exact, in a sense, in case the surface is the projective
plane.
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3
An efficient way of determining the value k in Theorem 1.1 is described
in the next section.
2. DEFINITIONS
We follow the definitions as in chapter 3 of [22], with exceptions as noted.
A surface X is a compact connected Hausdorff space in which each point
has a neighbourhood homeomorphic to the complex plane C. Thus every
surface is homeomorphic to either a sphere with h handles, denoted Sh ,
for some nonnegative integer h, or a sphere with k crosscaps, denoted Nk
for some positive integer k. The surfaces S0 , S1 , N1 , N2 are respectively
called the sphere, the torus, the projective plane and the Klien bottle. A
homeomorphic image of C on X is called an open disc. A homeomorphic
image of R on X is called an open arc For X ⊆ σ, we denote by X, int(X)
and δX, the (topological) closure, interior and boundary of X. A (closed)
bordered surface with k holes is the topological space X(k) := X − ∪ki=1 Di
where X is a surface and D1 , . . . Dk are open discs in X whose closures are
(1)
(2)
pairwise disjoint. Thus S0 is a closed disc. The bordered surfaces S0
(1)
and N1 are respectively called the closed cylinder and Möbius band .
An embedding of a graph G on a bordered surface X is function φ on
V (G) ∪ E(G) mapping vertices to distinct points on X and edges to disjoint
open arcs in X such that for any uv ∈ E(G) we have δφ(uv) = {φ(u), φ(v)},
and such that each arcwise connected component of X − φ(V ∪ E) is an
open disc (called a face). Such a pair (G, φ) is called an embedded graph
or an X-embedded graph. We often write G for (G, φ), and |G| for X when
the embedding is understood. Accordingly, we often identify the vertices
and edges of G with their images under φ. We denote the set of faces
of an embedded graph G by F (G). An embedded graph G = (V, E, F )
can be combinatorially described by specifying the graph (V, E) and, for
each f ∈ F (G), a closed walk Wf in (V, E) specifying the boundary of
f . The walk Wf is called a boundary walk of f . It is easy to see that
a triple (V, E, {Wf : f ∈ F }) specifies an embedded graph if and only if
each e ∈ E appears exactly twice among all the boundary walks, and for
each v ∈ V there is a cyclic list (e0 , e1 , . . . , ed = e0 ) of incident edges such
that ei and ei+1 appear as consecutive edges in some boundary walk for
0 ≤ i < d. The cyclic list is called the rotation at v, and d is the degree of
v. Any e ∈ E appears exactly twice among all rotations, and e may appear
twice in a single rotation or a single boundary walk. For f ∈ F , any edge
appearing in Wf is incident with f . The length of f is the length of Wf .
By interchanging the roles of V and F in G = (V, E, F ) in an obvious way,
one obtains a new embedded graph G∗ = (V ∗ , E ∗ , F ∗ ) = (F, E, V ) called
the surface dual of G. There is an obvious bijection between E and E ∗ .
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A walk in G∗ corresponds to a sequence of edges and faces of G which is
called a dual walk in G. A dual walk is simple it corresponds to a path or
circuit in G∗ .
If e ∈ E(G) bounds distinct faces, then an new embedded graph, denoted
G\e may be obtained by deleting e and identifying its two incident faces
in an obvious way. If e ∈ E(G) is not a loop, then a new embedded
graph, denoted G/e may be obtained by contracting e and identifying its
two incident vertices in an obvious way. Any embedded graph obtained
from G by successively deleting edges and contracting edges is called a
surface minor of G. A b-contraction of G is a contraction G/e such that
e is incident to a vertex of degree one. Any embedded graph obtained
from G by successively deleting edges and applying b-contractions is called
a surface subgraph of G. If G is an embedded graph, then the notation
H ⊆ G indicates that H is a surface subgraph of G. An embedded graph
G is even-faced if all its faces have even length. This is equivalent to saying
G∗ is Eulerian.
We briefly review some notions of surface topology. This development
essencially follows [27]. A curve on a bordered surface X is a continuous
function C : [0, 1] → X. The curve is closed if C(0) = C(1), open if
C(0) 6= C(1), and simple if C is injective on [0, 1). Let C, C 0 be closed
curves on X. A transformation from C to C 0 on X is a continuous function
T : [0, 1] × [0, 1] → X such that for each t ∈ [0, 1] the projection x 7→ T (x, t)
is a closed curve on X, and in particular C(x) = T (x, 0), and C 0 (x) =
T (x, 1). If such a transformation exists, then C and C 0 are freely homotopic.
This is an equivalence relation whose equivalence classes are called free
homotopy classes. A curve C is contractible on X, if C is free-homotopy
equivalent to a constant function. If two curves C, D : [0, 1] → X satisfy
C(1) = D(0), then their composition C · D is the curve is defined by
(C · D)(x) =
C(2x)
if 0 ≤ x ≤ 1/2
D(2x − 1) if 1/2 < x ≤ 1.
The inverse of C is defined by C −1 : x 7→ C(1−x). These definitions extend
without difficulty to the free homotopy classes of X. The free homotopy
classes of a bordered surface X forms a group under composition called the
(first) fundamental group π1 (X) of X. The fundamental groups of S0 and C
are trivial. The group π1 (N1 ) is the two-element group, whose nonidentity
element corresponds to the noncontractible curves in N1 . The fundamental
group of a cylinder is the free abelian group generated by a simple closed
curve along one boundary component.
A covering map is a surjective map f : X → Y preserving open sets which
restricts to a homeomorphism in a neighbourhood of each x ∈ X. Each
preimage f −1 (y) is a discrete subset of X whose cardinality is independent
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5
of X. We say X is simply connected if it has trivial fundamental group. A
universal cover of Y is a simply connected space X for which there exists
a universal covering map f : X → Y . A universal cover is unique up to
homeomorphism.
Two curves C, C 0 are homotopic if there exists a transformation T :
[0, 1] × [0, 1] → X such that T (0, t) and T (1, t) are constant functions of
t. This is an equivalence relation, where equivalent paths have common
endpoints, and equalence classes are called homotopy classes. Fix a point
p ∈ X, let Hom(p) denote the set of homotopy classes of curves C where
C(0) = p. Each λ ∈ Hom(p) consists of curves C with a common terminus
C(1). The universal covering surface of a bordered surface (with respect to
p) is the space U (X) whose point set is Hom(p). A subset T ⊆ X0 is open
if for any λ ∈ T , there exists an open disc N in X containing the terminus
q of λ such that for each curve D in Hom(q) ∩ N , the composition C · D
belongs to T . It turns out that U (X) is essencially independent of p. The
continuous function φ : U (X) → X which maps any point λ to its terminus,
is called the projection of X. The preimage of q ∈ X under φ is a discrete
subset of U (X) called the fibre of q. In particular, U (N1 ) is the sphere, and
the covering map is 2 to 1. For h ≥ 1, U (Sh ) is homeomorphic to C.
For any simple closed dual walk ω = f0 , e1 , f2 , . . . , ek , f0 on an embedded
graph G, the union of the (open) 1-cells and 2-cells corresponding to edges
and faces of ω is a 2-manifold homeomorphic to either a cylinder or a
Möbius band without boundary. We say that ω is two-sided if it is a
cylinder, and one-sided if it is a Möbius band. If |G| = P, then ω is twosided if and only if the circuit in G∗ corresponding to ω is contractible.
Let C 0 (x) = e2πix be the curve in C bounding the open set D 0 = {z ∈
C : |z| < 1}. Let X be a 2-manifold. A set D ⊆ X is an open disc if
there is a homeomorphism from D 0 to D. Let D ⊆ X be an open disc. A
curve C in X is called a boundary curve of D if C(x) = h(C 0 (x)) for some
continuous extension h of a homeomorphism h0 : D0 → D. Let X ⊆ X.
A closed curve C surrounds X on X, if there exists an open disc D ⊆ X
containing X such that there exists a transformation on X − D from C to a
boundary curve of D. We often identify a walk W on an embedded graph
G with a corresponding curve CW on |G|. Each edge in W corresponds to
a subinterval of [0, 1], which is mapped injectively by CW onto the arc in
|G| representing that edge. A closed walk W on G is a boundary walk of
an open disc D ⊆ |G| if it corresponds to a boundary curve CW of D. If
W is a boundary walk of an open disc D in X, then it is possible to form
a surface subgraph of G by successively deleting or b-contracting all edges
whose interiors are contained in D. We denote this surface subgraph by
G − D. We have D ∈ F (G − D).
If |G| 6= SS, then a closed walk W in G is the boundary walk of at most
one open disc D ⊆ X. If D is the unique open disc with boundary walk
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W , then D is called the interior of W , written D = int(W ). For example,
int(C) is well defined for any contractible circuit in G, provided |G| 6= SS.
If int(W ) exists, then we can not conclude that W is a circuit of G. For
example, if G consists of a single vertex with a single noncontractible loop
on the projective plane, then the unique face of G has length 2 whose
boundary a walk traverses that loop exactly twice.
The edge-width of G, written edgewidth(G), is the length of a shortest noncontractible closed walk in G. Let maxfl(G) denote the maximum
length of a face of G. Let minmaxfl(G) denote the minimum possible value
of maxfl(H) among the surface subgraphs H of G. That is,
minmaxfl(G) := min max length(f ).
H⊆G f ∈F (H)
(1)
This is the invariant of interest, according to Theorem 1.1.
Let f be a face of an embedded graph G. We denote by W(f ) = WG (f )
the set of closed walks in G which surround f . A family F of subsets of a
set X is laminar if, for any x, y ∈ F, at least one of the sets x − y, y − x,
x ∩ y is empty.
3. PRELIMINARIES
Here we derive an alternative formulation of minmaxfl(G) which is better
suited for proofs and for its efficient computation. The proofs of the following propositions have been omitted since they are not difficult, although
they are somewhat tedious to describe. [[Should I do it?]]
Proposition 3.1. Let H be a S0 -embedded graph, let v∞ ∈ V (H). For
each v ∈ V (H) − {v∞ } let Xv ⊆ V (H) be a minimal set of vertices such
that v ∈ Xv and δXv is a minimum size edge cut separating v from v∞ .
Then {Xv : v ∈ V (H) − {v∞ }} is a laminar family.
Proof. Let u, v ∈ V (H) − {v∞ }. Suppose that each of the sets U :=
Xu − Xv V := Xv − Xu W := Xu ∩ Xv is nonempty. Then the set
Z := V (H) − U − V − W is nonempty since v∞ ∈ Z. By submodularity of
edge-cut sizes we have the following.
|δU | + |δV | ≤ |δXu | + |δXv |
|δW | + |δZ| ≤ |δXu | + |δXv |
(2)
(3)
Suppose that u ∈ U and v ∈ V . By choice of Xu and Xv we have
|δU | > |δXu | and |δV | > |δXv |. These inequalities contradict (2), so either
u or v belongs to W .
We may assume u ∈ W . By choice of Xu we have that |δZ| ≥ |δXu |.
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7
If v ∈ W , then by choice of Xv we have |δW | > |δXv |, and the last
two inequalities contradict (3). Thus v ∈ V and u ∈ W . By choice
of Xv and Xu we have |δZ| ≥ |δXv | and |δW | > |δXu |. Adding these
inequalities, we get a contradiction to (3) which proves the proposition.
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Proposition 3.2. Let G be an embedded graph where |G| is not the
sphere. For each f ∈ F (G), let Wf be a shortest walk in W(f ) which
surrounds as few faces of G as possible. Then {int(Wf ) | f ∈ F (G)} is
a laminar family of open discs in |G|. Additionally, if f, f 0 ∈ F (G) and
int(Wf ) ⊆ int(Wf 0 ), then length(Wf ) ≤ length(Wf 0 ).
Proof. Let G0 be the universal covering graph of G. If |G| = N1 then
|G0 | = S0 and G0 is a 2-fold cover of G. Otherwise |G0 | is homeomorphic to
C (by a theorem of Poincaré), and G0 is an infinite graph. Let π : |G0 | → |G|
be the projection function. For each f 0 ∈ F (G0 ), let Uf 0 be the unique
closed walk in G0 surrounding f 0 such that π(Uf 0 ) = Wπ(f 0 ) . [[We assume
0
0
that π is injective on such int(f 0 )]] It suffices to prove that {int(Uf 0 ) | f ∈ F (G )}
0
is a laminar family of open discs in |G |.
We consider the case |G| = N1 . Here π is 2 to 1 and |G0 | = S0 . The
two preimages of f ∈ F (G) are said to be opposite on G0 . Let H be the
surface dual of G0 . Suppose {int(Uf 0 ), int(Ug0 )} is not laminar for some
f 0 , g 0 ∈ V (H). The vertices of H opposite to f 0 and g 0 on H are connected
by a path whose edges are disjoint from int(Uf 0 ) ∪ int(Ug0 ). By contracting
these edges, we get a new surface subgraph H 0 in which f∞ denotes the
new vertex corresponding to the contracted path. Since Wf surrounds f ,
deleting the edge set E(Uf 0 ) from H 0 separates f 0 from f∞ . By hypothesis,
E(Uf 0 ) is a minimal f 0 , f∞ -separating edge cut in H 0 which has the form
δXf for some minimal subset Xf 0 of V (H 0 ) where f 0 ∈ Xf 0 . A similar
statement holds for the edges of Ug0 . The sets Xf 0 and Xg0 are not laminar.
this contradicts the statement of Proposition 3.1, and proves the case that
|G| is the projective plane.
Now suppose |G| is not the projective plane. Let H be the infinite dual
graph of G0 , and suppose that {int(Uf 0 ), int(Ug0 )} is not laminar for some
f 0 , g 0 ∈ V (H).
YIKES!
Let F ⊆ F (G0 ) be such that for each f ∈ F (G) there
S exists a unique
0
f ∈ F such that π(int(f 00 )) = int(f ), and the closure of {int(f 0 ) | f 0 ∈ F}
is a closed disc in |G0 |. It suffices to prove that {int(Uf 0 ) | f 0 ∈ F} is a
laminar family of open discs in |G0 |.
Select f 0 ∈ F (G0 ) such that f 0 is an arcwise connected component of
−1
π (f ).
Proof. OLD ATTEMPT: Let f, f 0 ∈ F (G). Let W ∈ W(f ), W 0 ∈
W(f 0 ) be chosen as in the hypothesis. Let X = int(W ), X 0 = int(W 0 ).
Suppose each of X − X 0 , X 0 − X, X ∩ X 0 is nonempty. Each connected
component of X − W 0 is an open disc. At least one of these discs, say D,
has a boundary walk δD which consists of a subwalk of S of W , followed
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by a subwalk S 0 of W 0 . We consider two cases, depending on whether
D ⊆ X − X 0 or D ⊆ X ∩ X 0 .
Case 1 Suppose D ⊆ X − X 0 .
We claim that int(f ) ⊆ D. Suppose not. Let W −S+S 0 denote the closed
walk obtained by replacing S by S 0 in W . Then W −S +S 0 belongs to W(f )
and surrounds strictly fewer faces than does W . By choice of W , W −S +S 0
is strictly longer than W , so length(S 0 ) > length(S). Now W 0 − S 0 + S
belongs to W(f 0 ) and is stictly shorter than W 0 . This contradicts the choice
of W 0 and proves the claim.
We have f ⊆ D ⊂ X. Therefore S + S 0 ∈ W(f ), so the choice of W
implies that
length(W − S) < length(S 0 ).
(4)
Suppose f 0 ⊆ X 0 − X. Then W 0 − S 0 + (W − S) ∈ W(f 0 ) and is strictly
shorter than W 0 by (4), contradicting the choice of W 0 . Therefore f 0 ⊆
X 0 ∩ X. It follows that W − S + S 0 ∈ W(f 0 ). By choice of W 0 we have
length(W 0 − S 0 ) ≤ length(W − S). On the other hand, W 0 − S 0 + S ∈ W(f )
so, by choice of W , we have length(W − S) ≤ length(W 0 − S 0 ). These two
inequalities imply
length(W − S) = length(W 0 − S 0 ).
We have that W 0 − S 0 + S ∈ W(f 0 ) so, by choice of W 0 ,
length(S 0 ) ≤ length(S).
Case
2
Suppose
now
D
⊆
X
∩
X 0.
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Lemma 3.3. Let f be a face of a graph G embedded on C or on P. Then
the length of a shortest circuit in W(F ) can be computed in polynomial
time.
Proof. Suppose G is embedded on C. Since C − int(f ) is a topological
cylinder, its homotopy group is the free cyclic group. A closed walk in
C − Int(f ) belongs to W(f ) if and only if its homotopy class generates
that group. If a vertex appears twice in a closed walk W ∈ W(f ), then, W
has a proper subwalk whose homotopy class also generates the homotopy
group. It follows that every shortest walk in W is a simple circuit in G.
Let f ∗ and g ∗ denote the vertices of G∗ corresponding to f and the infinite
face of G. Then the simple circuits in G correspond to minimal f ∗ , g ∗ separating edge cuts in G∗ . It follows that a shortest circuit in W(f ) can
be found by generating G∗ and applying a max-flow min-cut algortithm to
find a minimum weight f ∗ , g ∗ -separating edge cut G∗ . This is a polynomial
time procedure.
Suppose now that G is enbedded in the projective plane P. Let π : SS →
P a universal 2-cover, and let G0 be embedded on SS such that π(G0 ) = G.
Let f 0 , f 00 be the two face of G0 in the preimage of G0 . Then for any closed
walk W in G, we have W ∈ W(f ) if and only if the preimage π(W ) is a
circuit of unit homotopy class in SS − int(f 0 ) − int(f 00 ). Since G0 can be
constructed in polynomial time, we are done by the previous paragraph.
Remark 3. 1. It is interesting to ask whether Lemma 3.3 can
be extended to graphs G embedded on other surfaces X. If X is not
the plane or projective plane, then there exists an infinite graph G0
embedded on C and a universal cover π : C → X which maps G0 to
G. One is tempted to compute a finite surface subgraph G00 ⊆ G0
consisting of all faces, edges and vertices of G0 that are within some
distance d of a fixed face f 0 in φ−1 (f ). The idea is to compute a
shortest closed walk in WG00 (f 0 ) as in the above proof. The problem
is to polynomially bound the size of G00 . If X is the torus or Klein
bottle, then |E| grows with the square of d, and such a scheme
can be made to work, although we omit details here. When X
has negative characteristic, then the size of G0 grows exponentically
with d. Thus we ask the following:
Given a (combinatorially described) graph G embedded on some 2-manifold,
and a face f ∈ F (G), can a shortest closed walk surrounding f be computed in
polynomial time?
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Lemma 3.4. For any graph G embedded on a surface X we have
minmaxfl(G) =
max
min
f ∈F (G) W ∈W(f )
length(W ).
(5)
Proof. Let k denote the right-hand side of (5). Let H0 be a surface
subgraph of G which is optimal for (1). For f ∈ F (G), let Wf ∈ W(f ) be a
closed walk which is optimal for the right hand side of (5). Every f ∈ F (G)
is surrounded by some closed walk W f which is the boundary walk of some
face of H0 . We have length(Wf ) ≤ length(W f ) ≤ minmaxfl(G). This
inequality holds for every f ∈ F (G), so k ≤ minmaxfl(G).
For each f ∈ F (G) we select Wf ∈ W(G) as in the previous paragraph,
but, subject to this, Wf is chosen to surround as few faces of G as possible.
By Proposition 3.1, {int(Wf ) | f ∈ F (G)} is a laminar family of open
discs in X. We construct a surface subgraph H0 ⊆ G by deleting (or bcontracting) any edge of G whose interior is contained in int(Wf ) for some
f ∈ F (G). Let W0 be the boundary walk of a longest face of H0 . By
Proposition 3.1 we have W0 = Wf0 for some f0 ∈ F (G), and the face f0
achieves the maximum on the right-hand side of (5). We have
k = length(Wf0 ) = max length(f ) ≥ minmaxfl(G).
f ∈F (H0 )
We have proven equation (5).
For some faces f ∈ F (G), the closed walks W ∈ W(f ) appearing on the
right hand side of (5) might not be simple circuits in G. However, we may
further restrict the set W(f ) in case |G| is the projective plane. Let C(f )
denote the set of walks in W(f ) which are simple circuits. It is possible
that C(f ) is the empty set. We define the point-girth of an embedded graph
G as follows.
pointgirth(G) := max
min length(C)
f ∈F (G) C∈C(f )
We define pointgirth(G) = ∞ in case C(f ) = ∅ for some f ∈ F (G).
Lemma 3.5. For any graph G embedded on the projective plane, we have
minmaxfl(G) = min { 2 edgewidth(G), pointgirth(G) }
(6)
Proof. Since |G| is the projective plane, any noncontractible closed
walk of minimum length is a circuit which induces a surface subgraph of
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G having exactly one face of length 2g, where g = edgewidth(G). Thus
minmaxfl(G) ≤ 2g. Observing that C(f ) ⊆ W(f ) and equation (5), we
have that minmaxfl(G) is at most the right hand side of (6).
Suppose that f0 ∈ F (G) achieves the maximum in (5). Let W0 ∈ W(f0 )
have shortest possible length. Suppose that W0 is not a circuit. We aim
to show that length(W0 ) ≥ 2g. Let x be a vertex occurring at least twice
on W0 . Let then W0 is the concatenation of two nonempty closed subwalks
W1 , W2 which begin and end at x. By Proposition ??, int(W0 ) is the boundary walk of a unique open disc in X. Therefore no proper subwalk of W0
is contractible on X. [[This may be vague]] In particular, W1 and W2 each
have length at least g. It follows that length(W0 ) ≥ 2g. This completes the
proof.
We will use a formula for χc (G) which is based on orientations. An
orientation of a graph G is a directed graph H obtained by assigning one
of two possible directions to each edge of G. A walk in a directed graph H
is a walk in the underlying undirected graph G. A circuit in H is a simple
closed walk in H. Suppose W has a forward edges and b backward edges,
where W has length a + b. Note that an edge appearing more than once in
W may contribute more than once to a and b. We define the imbalance of
W in H to be
a+b
imbalH (W ) ≥
min(a, b)
We allow the value imbalH (W ) = ∞ if one of a, b equals zero.
It was observed by Minty [20] that χ(G) can be expressed in terms of
circuit imbalances. The following analogous result for χc (G) appears in [11].
Lemma 3.6. For any graph G we have
χc (G) = min max imbalH (C),
H
C
where H ranges over the set of orientations of G, and C ranges over the
set of circuits in H.
It is not difficult to see that this formula remains valid if we extend the
set of closed walks over which C ranges. That is, for any graph G we have
χc (G) = min max imbalH (W ),
H
W
(7)
where H ranges over the set of orientations of G, and W ranges over the
set of closed walks in H.
The following is a special case of the main result of [6]. We provide here
a short self-contained proof of this special case.
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13
Lemma 3.7. Suppose that G is a nonbipartite even-faced graph embedded
on the projective plane. Then
χc (G) ≥ 2 +
2
r−1
Where the 2r = maxfl(G).
Proof. An orientation of G is face-balanced if every face boundary walk
in G has exactly half its edges oriented in each direction. We claim that G
does not have a face-balanced orientation. Suppose H is a face-balanced
orientation of G. Let e be an edge incident with distinct faces of H. Then
H\e is a face-balanced orientation of G\e. A similar statement holds for
any b-contraction H/e of H. Thus, every surface subgraph of H is facebalanced. Let C0 be an odd circuit in H. Then imbalH (C0 ) > 2. Furthermore C0 is a noncontractible circuit in H which induces a surface subgraph
H0 of H. A boundary walk W0 of the unique face of H0 consists of two
traversals of C0 , so imbalH (W0 ) = imbalH (C0 ) > 2. This contradiction
proves the claim.
Therefore for any orientation H of G, there exists a face f such that
its boundary walk Wf satisfies imbalH (Wf ) ≥ 2s/(s − 1), where 2s =
length(f ). The result follows from (7) and the fact s ≤ k.
4. PROOF OF MAIN RESULT
Let G be an even-faced projective plane graph. We may assume G loopless and nonbipartite. Deleting an edge of G preserves the even-face property, so minmaxfl(G) is an even integer. A circuit in G has odd length if
and only if it is noncontractible. Thus edgewidth(G) is odd and greater
than two, and pointgirth(G) is even and positive, if not infinite. Let r, s, t
be positive integers such that
minmaxfl(G) = 2r,
pointgirth(G) = 2s,
edgewidth(G) = 2t + 1.
We permit the value s = ∞.
If s = ∞, then let f0 be any face of G. Otherwise, let f0 ∈ F (G) be such
that every closed walk surrounding G has length at least 2s.
Since |G| is the projective plane, the face f0 is the first and last face in
some one-sided simple closed dual walk ω = f0 , e1 , f1 . . . , ek , f0 in G. (For
example, we may define ω to correspond to a shortest noncontractible walk
through f0 in G∗ .)
We label the ends of each ei ∈ E(ω) with xi , yi so that the labels occur
in the order x1 , x2 , . . . , xk , y1 , y2 , . . . , yk along the boundary of the
14
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Möbius band associated with ω. We point out that it is possible for one
vertex to receive several labels in this scheme. Since G is loopless, the
surface minor G0 = G/{e1 , . . . , ek } exists. The odd-length circuits in G are
exactly the noncontractible circuits in G, and such circuits contain an odd
number of the edges in ω. It follows that G0 is bipartite. By Lemma 3.6,
there exists an edge-reorientation H 0 of G0 such that every circuit in H 0
has imbalance 2.
We can extend H 0 to an edge-reorientation H of G by directing each
ei ∈ E(ω) from xi to yi . It is not difficult to verify that every circuit C
satisfies

 ±2 if C surrounds f0
a − b = ±1 if C is not contractible

0 otherwise.
where C contains a forward edges and b backward edges of H. Therefore
 2m

 m−1 if C surrounds f0 , and has length 2m
2`+1
imbalH (C) =
if C is not contractible, and has length 2` + 1
`


2 otherwise.
By choice of f0 and the hypothesis, we have 2m ≥ 2s and 2` + 1 ≥ 2t + 1.
Therefore every circuit C has imbalance at most
2
2
2s 2t + 1
=2+
,
=2+
.
max
s−1
t
min{s − 1, 2t}
r−1
The final equation comes from Lemma 3.5. (If s = ∞, then no circuit
surrounds f0 , and the above equation is still valid provided we interpret
2s
s−1 = 2.) It follows from Lemma 3.6 that χc (G) ≤ 2 + 2/(r − 1).
It remains to prove the opposite inequality. Let H be a surface subgraph
of G such that maxfl(H) = minmaxfl(G) = 2r. Then χc (G) ≥ χc (H).
However, we have χc (H) ≥ 2/(r − 1) by Lemma 3.7 This completes the
proof.
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