R. China Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial. Key words: totally cyclic orientation, Eulerian-equivalence class, integral flow polynomial, modular flow polynomial, isomorphism AMS Classification: 05C70, 05C30 In this note we present an answer to an open problem proposed by Beck and Zaslavsky [1] in the decomposition theory of flow polynomials, by showing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial. ∗ Corresponding author

Arthur L. B. Yang

Beifang Chen

AbstractUsing the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial

Chen, Beifang

Yang, Arthur L.B.

Elsevier - Publisher Connector 

Discrete Mathematics 309 (2009) 1708–1710
www.elsevier.com/locate/disc
Note
A note on flow polynomials of graphs
Beifang Chena, Arthur L.B. Yangb
a Department of Mathematics, Hong Kong University of Sciences and Technology, Clear Water Bay, Kowloon, Hong Kong
b Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China
Received 9 August 2007; received in revised form 9 November 2007; accepted 24 January 2008
Available online 12 March 2008
Abstract
Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by
providing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial.
c© 2008 Elsevier B.V. All rights reserved.
Keywords: Totally cyclic orientation; Eulerian-equivalence class; Integral flow polynomial; Modular flow polynomial; Isomorphism
In this note we present an answer to an open problem proposed by Beck and Zaslavsky [1] in the decomposition
theory of flow polynomials, by showing a general situation in which the integral flow polynomial is a multiple of the
modular flow polynomial.
Let us start with reviewing some definitions and notations. By a graph G = (V (G), E(G)), we mean that G has
finite vertex set V (G) and edge set E(G). Loops and multiple edges are allowed. Assume that all graphs considered
here are connected. Each edge e ∈ E(G) is incident with two vertices u, v ∈ V (G), and it can be assigned a direction
either from u to v or from v to u, but not both. In particular, a loop has two directions from a vertex to itself. If each
edge of G has a direction, we say that G is oriented. Let O(G) denote the set of all orientations of G.
Fix an orientation ε ∈ O(G). By a circle of G we mean a 2-regular connected subgraph in G. A directed cycle is
a circle in which all edges have a consistent direction with respect to ε. We say that ε is a totally cyclic orientation if
every edge belongs to a directed cycle. Denote the set of totally cyclic orientations by CO(G). Given two orientations
ε1, ε2 ∈ O(G), let E(ε1 6= ε2) denote the subset of E(G) composed of the edges having opposite directions with
respect to ε1 and ε2. We say that ε1, ε2 are Eulerian-equivalent if E(ε1 6= ε2) can be written as a disjoint union
of directed cycles. As shown in [3,4], Eulerian-equivalence relation is indeed an equivalence relation on O(G) and
induces an equivalence relation on CO(G). Let [ε] be the Eulerian-equivalence class of ε. Let [CO(G)] be the set of
all Eulerian-equivalence classes in CO(G).
For a fixed orientation ε ∈ O(G) and a given vertex v ∈ V (G), let E+(v, ε) be the set of edges taking v as the
head and E−(v, ε) the set of edges taking v as the tail. A nowhere-zero flow of G is a function f : E(G)→ A such
that ∑
e∈E+(v,ε)
f (e) =
∑
e∈E−(v,ε)
f (e), ∀v ∈ V (G)
E-mail addresses: mabfchen@ust.hk (B. Chen), yang@nankai.edu.cn (A.L.B. Yang).
0012-365X/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2008.01.050
B. Chen, A.L.B. Yang / Discrete Mathematics 309 (2009) 1708–1710 1709
holds, where A is an abelian group and f never takes the value 0. Taking A = Z, an integral nowhere-zero flow is
called a nowhere-zero k-flow if | f (e)| < k for every edge e ∈ E(G). It was known that both the number of nowhere-
zero flows with values in a finite abelian group and the number of nowhere-zero k-flows are independent of the chosen
orientation and the actual group structure [3,4,6,7]. The former one is a polynomial function of the order of the finite
abelian group A, and is called the modular flow polynomial, denoted φ(G, k). The latter one is a polynomial function
of k, and is called the integral flow polynomial, denoted φZ(G, k). For each orientation ε ∈ O(G), the number
of nowhere-zero integral flows with values in {0, 1, . . . , k − 1} is also a polynomial function of k [3,4], denoted
φZ(G, ε, k).
Lemma 1 ([3,4]). If ε and ε′ are two Eulerian-equivalent orientations, then
φZ(G, ε, k) = φZ(G, ε′, k). (1)
Beck and Zaslavsky noticed that both φZ(3K2, x) (3K2 is the graph of 3 parallel links) and φZ(K4, x) (K4 is
the complete graph of 4 vertices) have integral coefficients, and moreover φZ(3K2, x) = 3φ(3K2, x), φZ(K4, x) =
4φ(K4, x); see [1, Example 3.4]. Then, they proposed the following problem [1, Problem 3.5]:
Is there any general reason why for 3K2 and K4 both of the integral flow polynomials have integral coefficients
and the integral flow polynomial is a multiple of the modular polynomial?
Now we can answer the above problem. Note that the modular flow polynomials always have integral
coefficients [1,3]. Therefore, we focus on the reason why φZ(G, x) is a multiple of φ(G, x) for some graph G. In
fact, the answer is implied in the following theorem due to Kochol [4, Equations (1) and (2)] and Chen and Stanley [3,
Theorems 4.4 and 5.6], which shows that the modular and integral flow polynomials admit a nice decomposition.
Proposition 2 ([3,4]). Given a graph G, let φ(G, x) be the modular flow polynomial and φZ(G, x) the integral flow
polynomial. Then
φ(G, x) =
∑
[ε]∈[CO(G)]
φZ(G, ε, x) (2)
φZ(G, x) =
∑
ε∈CO(G)
φZ(G, ε, x). (3)
Note that similar results hold for integral and modular tension polynomials of graphs, see [2,5]. Our main result is
an immediate consequence of Lemma 1 and Proposition 2.
Theorem 3. If each equivalence class of [CO(G)] has the same number of totally cyclic orientations, say m, then we
have
φZ(G, x) = mφ(G, x). (4)
Before answering the problem of Beck and Zaslavsky, let us recall the definition of isomorphism between two
directed multigraphs. Suppose that D = (V, E) and D′ = (V ′, E ′) are directed multigraphs. We say that D is
isomorphic to D′ if there is a bijection θ : V → V ′ such that for all vertices u, v in V the number of edges from u
to v in D is the same as the number of edges from θ(u) to θ(v) in D′. Then θ is called an isomorphism from D to
D′. Given a graph G, note that each orientation ε ∈ O(G) determines a unique directed graph Dε. We say that two
orientations ε1, ε2 ∈ O(G) are isomorphic if Dε1 and Dε2 are isomorphic to each other. By Theorem 3, we obtain the
following result.
Corollary 4. If G is a graph such that any two non-Eulerian equivalent orientations in CO(G) are isomorphic, then
the integral flow polynomial φZ(G, x) has integral coefficients and is a multiple of the modular flow polynomial
φ(G, x).
Proof. If [CO(G)] has only one Eulerian-equivalence class, then the result clearly holds. Otherwise, suppose that ε1
and ε2 are any two totally cyclic orientations which are not Eulerian-equivalent, and we will show that [ε1] and [ε2]
have the same number of totally cyclic orientations. Let θ be the isomorphism from Dε1 to Dε2 . For any orientation
1710 B. Chen, A.L.B. Yang / Discrete Mathematics 309 (2009) 1708–1710
ε′1 in [ε1], the edge set E(ε1 6= ε′1) can be written as a disjoint union of directed cycles with respect to ε1. Since ε1
and ε2 are isomorphic, then the edge set E(ε1 6= ε′1) can also be written as a disjoint union of directed cycles with
respect to ε2. For the orientation ε2, by reversing the direction of the edges in E(ε1 6= ε′1) and keeping the direction
of other edges, we obtain an orientation ε′2 which obviously belongs to [ε2]. Note that the orientation ε′2 is uniquely
determined by ε′1 when fixing ε1, ε′1 and the isomorphism from Dε1 to Dε2 . This implies that the cardinality of [ε1] is
less than or equal to that of [ε2]. Similarly, we can prove that the cardinality of [ε2] is less than or equal to that of [ε1].
Therefore, each Eulerian-equivalence class of [CO(G)] has the same number of totally cyclic orientations. Applying
Theorem 3, we complete the proof. 
In particular, we have the following conclusion.
Corollary 5. If G is 3K2 or K4, then any two totally cyclic orientations of CO(G) are isomorphic. Therefore, in both
cases the integral flow polynomial is a multiple of the modular flow polynomial.
Proof. Let us first consider graph 3K2 with the vertex set {v1, v2}. For any totally cyclic orientation ε ∈ CO(3K2), we
have either |E+(v1, ε)| = 2 or |E+(v1, ε)| = 1. Given any two totally cyclic orientations ε1 and ε2, if |E+(v1, ε1)| =
|E+(v1, ε2)| then the identity map is an isomorphism between Dε1 and Dε2 . If |E+(v1, ε1)| 6= |E+(v1, ε2)|, then we
take the bijection θ given by θ(v1) = v2 and θ(v2) = v1 as the desired isomorphism.
For any totally cyclic orientation ε ∈ CO(K4), the equality∑
v∈V (K4)
|E+(v, ε)| =
∑
v∈V (K4)
|E−(v, ε)|
forces that there are exactly two vertices, say v1, v2, such that |E+(v1, ε)| = |E+(v2, ε)| = 2, and exactly two
vertices, say v3, v4, such that |E+(v3, ε)| = |E+(v4, ε)| = 1. Without loss of generality, we may assume that for ε
the edge incident with {v1, v2} is directed from v2 to v1 and the edge incident with {v2, v3} is directed from v3 to v2.
In this case, the directions of the remained edges are uniquely determined. For any two orientations ε1, ε2 ∈ CO(K4),
we label the vertices of Dε1 and Dε2 as above. Suppose that V (Dε1) = {v1, v2, v3, v4} and V (Dε2) = {v′1, v′2, v′3, v′4}.
Then the bijection θ with θ(vi ) = v′i is clearly an isomorphism between Dε1 and Dε2 . 
Theorem 3, Corollaries 4 and 5 actually present an answer to the problem of Beck and Zaslavsky. For the graph
3K2, there are 2 Eulerian-equivalence classes in [CO(3K2)], and each class has 3 totally cyclic orientations. Therefore,
φZ(3K2, x) = 3φ(3K2, x). For the graph K4, there are 6 Eulerian-equivalence classes in [CO(K4)], and each class
has 4 totally cyclic orientations. Therefore, φZ(K4, x) = 4φ(K4, x). However, there exists some graph G for which
the condition of Corollary 4 is not satisfied but its integral flow polynomial φZ(G, x) is still a multiple of φ(G, x).
For such a graph, the reader may consult the graph K4 minus one edge.
Acknowledgments
The research was supported by RGC Competitive Earmarked Research Grant 600506. The second author was also
supported by the 973 Project on Mathematical Mechanization, the PCSIRT Project of the Ministry of Education, the
Ministry of Science and Technology, and the National Science Foundation of China. We would like to thank Professor
Thomas Zaslavsky for his interest in this work and we are also grateful to two anonymous referees for their valuable
comments.
References
[1] M. Beck, T. Zaslavsky, The number of nowhere-zero flows on graphs and signed graphs, J. Combin. Theory Ser. B 96 (6) (2006) 901–918.
[2] B. Chen, Orientations, lattice polytopes, and group arrangements. I: Chromatic and tension polynomials of graphs, preprint.
[3] B. Chen, R.P. Stanley, Orientations, lattice polytopes, and group arrangements. II: Modular and integral flow polynomials of graphs, preprint.
[4] M. Kochol, Polynomials associated with nowhere-zero flows, J. Combin. Theory Ser. B 84 (2002) 260–269.
[5] M. Kochol, Tension polynomials of graphs, J. Graph Theory 40 (3) (2002) 137–146.
[6] G.-C. Rota, On the foundations of combinatorial theory. I, Theory of Mo¨bius functions, Z. Wahrscheinlichkeitstheorie 2 (1964) 340–368.
[7] W.T. Tutte, On the embedding of linear graphs in surfaces, Proc. London Math. Soc. 52 (2) (1950) 474–483.


A note on flow polynomials of graphs 

Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general Situation in which the integral How polynomial is a multiple of the modular flow polynomial. (C) 2008 Elsevier B.V. All rights reserved

Yang, Arthur L. B.

Hong Kong University of Science and Technology Institutional Repository

A note on flow polynomials of graphs

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