enumeration of graph coverings

YOKOHAMA MATHEMATICAL JOURNAL VOL. 47, 1999 SPECIAL ISSUE

ENUMERATION OF GRAPH COVERINGS* By

JAEUN LEE (Received February 22, 1999)

Abstract. Enumerative research is presently a major center of interest in topological graph theory, as in the work of Archdeacon et al [2], Gross and Furst [4], Hofmeister $[9]-[13]$ , Kwak and Lee $[20]-[24]$ , Mizuno and Sato $[25]-[28]$ and [32], Mohar [29], Mull, Rieper and White [30], Negami [31], etc. In this paper, we survey some results on the enumeration of graph coverings and introduce some unsolved problems.

Introduction be a connected finite simple graph with vertex set $V(G)$ and edge set . The neighborhood of a vertex $v\in V(G)$ , denoted by $N(v)$ , is the set of vertices adjacent to . We use $|X$ I for the cardinality of a set $X$ . The number $\beta(G)=|E(G)|-|V(G)|+1$ is equal to the number of independent cycles in $G$ and it is referred to as the Betti number of $G$ . Let $Aut(G)$ be the group of all automorphisms of $G$ . A graph is called a covenng of $G$ with projection if there is a surjection $p:V(\tilde{G})\rightarrow V(G)$ such that : is a bijection for all vertices $v\in V(G)$ and . We also say that the projection is an n-fold covering of $G$ if is $n- t\infty one$ . A covering is said to be regular if is a subgroup $A$ of the automorphism group of acting $G$ heely on such that the quotient graph is isomorphic to $G$ . The fiber of an edge or a vertex is its preimage under . Let be a group of automorphisms of the graph $G$ . Two coverings : $G,$ $i=1,2$ , are said to be isomorphic with respect to or -isomorphic if there exist a graph isomorphism : and a graph automorphism such that the diagram Let

$G$

$E(G)$

$v$

$\tilde{G}$

$p:\tilde{G}\rightarrow G$

$p|_{N(\tilde{v})}$

$N(\tilde{v})\rightarrow N(v)$


$\tilde{v}\in p^{-1}(v)$


$p$

$Aut(\tilde{G})$

$there\sim$

$\tilde{G}$

$\tilde{G}/\mathcal{A}$

$p$

$\Gamma$

$\tilde{G}_{i}\rightarrow$

$p_{i}$

$\Gamma$

$\Phi$

$\Gamma$

$\tilde{G}_{1}\rightarrow\tilde{G}_{2}$

’Supported by BSRI-1409 1991 Mathematics Subject Classffication: $05C10$ , 05C30 Key words and phrases: graph covering, enumeration

$\gamma\in\Gamma$

102

J. LEE $\tilde{G}_{1}\rightarrow^{\Phi}\tilde{G}_{2}$

$p_{1\downarrow}G\underline{\gamma}G\downarrow p_{2}$

commutes. Such a is called a covering isomorphism with respect to or isomorphism. Note that for any group of automorphisms of $G$ , the covering isomorphic relation with respect to on the coverings of $G$ is an equivalence relation. Every edge of a graph $G$ gives rise to a pair of oppositely directed edges. By $e^{-1}=vu$ , we mean the reverse edge to a directed edge $e=uv$ . We denote the set of directed edges of $G$ by $D(G)$ . Following Gross and Tucker [7], a permutation voltage assignment of $G$ is a function : $D(G)\rightarrow S_{n}$ with the $\phi(e^{-1})=\phi(e)^{-1}$ property that for each $e\in D(G)$ , where is the symmetric group on elements $\{1, \ldots, n\}$ . The permutation denved graph is defined as follows: $V(G^{\phi})=V(G)\times\{1, \ldots, n\}$ , and for each edge $e=uv\in D(G)$ and $j\in\{1, \ldots, n\}$ , let there be an edge $(e,j)$ in joining a vertex $(u,j)$ $(v, \phi(e)j)$ and . The first coordinate projection : , called the natural projection, is a covering. Let $A$ be a finite group. An ordinary voltage assignment (or, A-voltage assignment) of $G$ is a function : $D(G)\rightarrow A$ with the property that $\phi(e^{-1})=\phi(e)^{-1}$ for each $e\in D(G)$ . The values of are called voltages, and $A$ is called $A$ derived from the voltage group. The ordinary derived graph $D(G)\rightarrow A$ an ordinary voltage assignment : has as its vertex set and as its edge set $E(G)\times A$ , so that an edge of joins a vertex $(u,g)$ to $(v, \phi(e)g)$ for $e=uv\in D(G)$ and $g\in A$ . The first coordinate projection : , also called the natural projection, commutes with the left multiplication action of the and the right action of on the fibers, which is free and transitive, so that is an $|A|$ -fold regular covering, called simply an $\Phi$

$\Gamma$

$\Gamma-$

$\Gamma$

$\Gamma$

$\phi$

$\phi$

$S_{n}$

$G^{\phi}$

$n$

$D(G^{\phi})$

$p^{\phi}$

$G^{\phi}\rightarrow G$

$\phi$

$\phi$

$ G\times\phi$

$\phi$

$V(G)\times \mathcal{A}$

$G\times\phi \mathcal{A}$

$p_{\phi}$

$G\times\phi A\rightarrow G$

$\phi(e)$

$\mathcal{A}$

$p_{\phi}$

A-coven $ng$ .

In this paper, we mainly discuss the enumeration of the I-isomorphism classes of coverings of $G$ , where is the trivial automorphism group {1} of $G$ . From now on, we shall assume that the isomorphism classes of coverings stand for the I-isomorphism classes of them. Waller [34] studied the structures of double coverings of $G$ . After then, Hofmeister [9] enumerated the isomorphism classes of double coverings of $G$ . Kwak and Lee $[20, 23]$ enumerated the isomorphism classes of graph bundles and those of all finite-fold graph coverings. Hofmeister $[10, 13]$ independently enumerated the isomorphism classes of all finite-fold graph coverings. But the enumeration of the isomorphism classes of regular graph coverings has not been answered completely. As its partial answers, Hofmeister [9] enumerated the isomorphism classes of regular double coverings of $G$ , Kwak and $I$

ENUMERATION OF GRAPH COVERINGS

103

Lee [21], and Sato [32] did the same work for regular prime-fold coverings. Hong and Kwak [14] did it for regular four-fold coverings. Hofmeister [11] enumerated the isomorphism classes of -coverings. Recently, Kwak et al [18] enumerated the isomorphism classes of $A$ -coverings when $A$ is a finite abelian group or the dihedral group . Notice that there are also several interesting result on the enumeration of the -isomorphism classes of graph coverings for any nontrivial subgroup of $Aut(G)$ . For example, Hofmeister [9] enumerated the F-isomorphism classes of double coverings and Sato $[28, 32]$ did the same work for or $h$ -coverings. When fixes a spanning tree of $G$ , some results on the enumeration of the -isomorphism classes of graph coverings can be found in [15] and [20]. It can be also found in [3], [12], [26] and [27] that several generalized notion of graph coverings and some interesting results on the enumeration of the isomorphism classes of them. This paper is organized as follows. In section 1, we deal with some algebraic characterizations when given two coverings are isomorphic and derive some new enumeration formulas for the F-isomorphism classes of regular coverings. In section 2, we treat some explicit enumeration formulas for graph coverings. In section 3, we conclude with some problems. $\oplus_{\dot{\iota}=1}^{n}\mathbb{Z}_{p}$

$D_{n}$

$s$

$\Gamma$

$\Gamma$

$\mathbb{Z}_{2}\times \mathbb{Z}_{2^{-}}$

$\Gamma$

$\Gamma$

1.

Algebraic characterizations

Gross and Tucker [7] showed that every covering (resp. regular covering) $G$ of can be described by a permutation (resp. ordinary) voltage assignment. By the virtue of their work, we may enumerate the isomorphism classes of coverings of $G$ by enumerating some equivalence classes of voltage assignments of $G$ . $\tilde{G}$

THEOREM 1. ([20]) Let be a subgrvup of $Aut(G)$ and : $D(G)\rightarrow S_{n}$ permutation voltage assignments. Then two n-fold coverings and are isomorphic with respect to if and only if there exist a function $f:V(G)\rightarrow S_{n}$ and such that $\Gamma$

$\phi,$

$\psi$

$G^{\phi}$

$G^{\psi}$

$\Gamma$

$\gamma\in\Gamma$

$\psi(\gamma u\gamma v)=f(v)\phi(uv)f(u)^{-1}$

for each

$uv\in D(G)$

.

Notice that an equivalent form of Theorem 1 can also be found in [10]. For a finite group $A$ , let denote the set of all bijective functions on the underlying set $A$ and denote the left translation group of $A,$ is the group of left translations for $g\in A$ , where for all . Then Theorem 1 can be rephrased for regular coverings as follows: $S_{A}$

$\mathcal{L}_{\mathcal{A}}$

$\mathcal{L}_{g}$

$i.e.,$

$\mathcal{L}_{g}(g^{\prime})=gg^{\prime}$

$\mathcal{L}_{A}$

$g^{\prime}\in \mathcal{A}$

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J. LEE

THEOREM 2. ([15]) Let be a subgrvup of $Aut(G)$ and A-voltage assign$A$ and $A$ are isomorphic with ments of G. Then two A-coverings respect to if and only if there exist a graph automorphism and a function $f:V(G)\rightarrow S_{A}$ such that $\Gamma$

$\phi,$

$ G\times\phi$

$\psi$

$ G\times\psi$

$\Gamma$

$\gamma\in\Gamma$

$\mathcal{L}_{\psi(\gamma u\gamma v)}=f(v)\circ \mathcal{L}_{\phi(uv)}of(u)^{-1}$

for each $uv\in D(G)$ . A finite group $A$ is said to have the isomorphism extension property (IEP) if every isomorphism between any. two isomorphic subgroups and of can be extended to an automorphism of . For example, the cyclic group for any natural number , the dihedral group for odd $n\geq 3$ , and the direct sum of $m$ copies of for a prime number have the IEP. Moreover, the direct sum of two groups whose orders are relatively prime has the IEP if each direct summand has it. However, neither nor $n=even$ has the IEP. $\mathcal{B}_{1}$

$\mathcal{A}$

$\mathcal{B}_{2}$

$\mathcal{A}$

$\mathbb{Z}_{n}$

$n$

$D_{n}$

$\mathbb{Z}_{p}$

$p$

$D_{n},$

$\mathbb{Z}_{2}\oplus \mathbb{Z}_{4}$

THEOREM 3. ([15]) Let be a subgroup of $Aut(G)$ and -voltage assignments of G. If either $A$ has the IEP or both and derive connected covenngs, then the following are equivalent: $\Gamma$

$\phi,$

$\phi$

(a) Two

$A$

-coverings

$ G\times\phi$

and

$A$

$ G\times\psi$

(b) There exist a graph automorphism $Aut(A)$

and a function

$f$

:

$A$

$\psi A$

$\psi$

are isomorphic with respect to . , a gro up automorphism such that $\Gamma$

$\gamma\in\Gamma$

$V(G)\rightarrow A$

$\sigma\in$

$\psi(\gamma u\gamma v)=f(v)\sigma(\phi(uv))f(u)^{-1}$

for each $uv\in D(G)$ . Moreover, if $A$ is abelian, then each the following conditi on:

of these two conditions

(c) There exist a graph automorphism $Aut(A)$ ,

and a function

is also equivalent to

, a group automorphism such that

$\gamma\in\Gamma$

$f:V(G)\rightarrow A$

$\sigma\in$

$\psi(\gamma u\gamma v)=\sigma(\phi(uv))f(v)f(u)^{-1}$

for each

$uv\in D(G)$

.

Let be an abelian group and let $H^{1}(G;A)$ be the first cohomology group $G$ of . We define a $\Gamma\times Aut(A)$ action on by $\mathcal{A}$

$H^{1}(G;\mathcal{A})$

$(\gamma, \sigma)\cdot[\phi]=[(\gamma, \sigma)\cdot\phi]$

,

where

$((\gamma, \sigma)\cdot\phi)(uv)=\sigma(\phi(\gamma^{-1}u\gamma^{-1}v))$


105

$[\phi]\in H^{1}(G;A)$ , and $uv\in D(G)$ . If $A$ has the IEP, then, for $A$ are isomorphic by Theorem 3, two A-coverings and with respect to if and only if and are in the ame orbit of the action on . Hence, the number of the -isomorphism classes of the A-coverings of $G$ is equal to $|H^{1}(G;A)/(\Gamma\times Aut(A))|$ . For a finite group $A$ and a subgroup of $Aut(G)$ , let Isor (resp., $Isoc_{\Gamma}(G;A))$ denote the number of the F-isomorphism classes of A-coverings (resp., connected -coverings) of $G$ . Let (resp., ) denote the number of the -isomorphism classes of regular n-fold coverings (resp., connected regular n-fold coverings) of $G$ . When is the trivial group, they are simply denoted by $Isoc(G;A),$ $Iso^{R}(G;n)$ and ISOC $(G;n)$ . By using Theorem 3, Kwak et al [18] derived some formulas for enumerating the isomorphism classes of regular coverings. Now, we aim to derive some enumeration formulas for the -isomorphism classes of regular coverings. It comes from the definition of a regular covering that each component of a regular n-fold covering of $G$ is also a connected regular d-fold covering of $G$ for a divisor of , and any two components are isomorphic with respect to the trivial automorphism group of $G$ . Hence, the number of the -isomorphism classes of regular n-fold coverings with $n/d$ component is equal to the number of the -isomorphism classes of connected regular d-fold coverings. Moreover, each component of an A-covering of $G$ is I-isomorphic to a connected S-covering of $G$ for a subgroup of $A$ . Let $A$ and $B$ be two non-isomorphic groups such that . Then, by a method similar to the proof of Theorem 3, we can have that, for each subgroup of $Aut(G)$ , every connected A-covering of $G$ can not be -isomorphic to any B-covering of $G$ . Hence, we have the following theorems. $(\gamma, \sigma)\in\Gamma\times Aut(A),$

$G\times\phi \mathcal{A}$

$\Gamma$

$[\phi]$

$ G\times\psi$

$[\psi]$

$s$

$H^{1}(G;\mathcal{A})$

$\Gamma\times Aut(\mathcal{A})$

$\Gamma$

$\Gamma$

$(G;\mathcal{A})$

$Iso_{\Gamma}^{R}(G;n)$

$\mathcal{A}$

$Isoc_{\Gamma}^{R}(G;n)$

$\Gamma$

$\Gamma$

$Iso(G;\mathcal{A}),$

$\Gamma$

$d$

$n$

$\Gamma$

$\Gamma$

$s$

$S$

$|\mathcal{A}|=|\mathcal{B}|$

$\Gamma$

$\Gamma$

THEOREM 4. For any natural number $Iso_{\Gamma}^{R}(G;n)=\sum_{d|n}Isoc_{\Gamma}^{R}(G;d)$

and

$n$

and any subgrvup

$\Gamma$

of $Aut(G)$ ,

$Isoc_{\Gamma}^{R}(G;n)=\sum_{A}Isocr(G;A)$

where A runs over all representatives order .

of the

THEOREM 5. For any finite group

and any subgroup

isomorphism classes

,

of groups of

$n$

$\mathcal{A}$

$Iso_{\Gamma}(G;A)=\sum_{s}Isoc_{\Gamma}(G;S)$

$\mathcal{A}wheoeS$

$\Gamma$

of $Aut(G)$ ,

,

runs over all representatives of the isomorphism classes of subgrv ups of

Theorems 4 and 5 say that the computation of the number may be completed if one can compute the number Isocr $(G;A)$ for each group $A$ whose $Iso_{\Gamma}^{R}(G;n)$

J. LEE

106

order is a divisor of . In general, for a non-trivial subgroup of $Aut(G)$ , it is very hard to derive a formula for computing the number $Isoc_{\Gamma}(G;A)$ . The has been done for only few cases (see computation of the number $\Gamma=I$ , Kwak et al [18] computed this number as the table in Section 3). When follows. $\Gamma$

$n$

$Isoc_{\Gamma}(G;\mathcal{A})$

THEOREM 6. For any finite group Isoc

where

$\mathfrak{G}(A;n)=$

{

$(g_{1},$ $g_{2},$

$A$

,

$(G;A)=\frac{|\mathfrak{G}(\mathcal{A};\beta(G))|}{|Aut(A)|}$

$\ldots g_{n})\in \mathcal{A}^{n}$

:

THEOREM 7. For any two finite groups

Isoc

, generates

$\{g_{1},g_{2},$ $\ldots g_{n}\}$

$\mathcal{A}$

and

$\mathcal{B}$

with

$A$

$(|A|, |\mathcal{B}|)=1$

$(G;\mathcal{A}\oplus \mathcal{B})=Isoc(G;\mathcal{A})Isoc(G;\mathcal{B})$

}.

,

,

and $Iso(G;A\oplus \mathcal{B})=Iso(G;\mathcal{A})Iso(G;\mathcal{B})$

.

Note that, even though $Iso(G;A\oplus \mathcal{B})=Iso(G;\mathcal{A})Iso(G;\mathcal{B})$ for $(|A|, |\mathcal{B}|)=1$ , it does not hold that $Iso^{R}(G;mn)=Iso^{R}(G;m)Iso^{R}(G;n)$ for two relatively prime numbers $m$ and , because there may be a group of order $mn$ which can not be expressed as a direct sum of two groups of order $m$ and . For example, the dihedral group D3 of order 6 is not isomorphic to the direct sum of the two cyclic groups and . $n$

$n$

$\mathbb{Z}_{2}$

2.

$\mathbb{Z}_{3}$

Some explicit formulas

In this section, we introduce some known explicit enumeration formulas for the isomorphism classes of graph coverings. The enumeration of the isomorphism clas es of n-fold coverings of $G$ was completely done by Kwak and Lee [20], and independently Hofmeister [10]. $s$

THEOREM 8. ([10], [20]) The number n-fold covenngs of $G$ is $Iso(G;n)=\sum_{\ell_{\iota+2\ell_{2}+\cdots+n\ell_{n}=n}}(\ell_{1} !

$Iso(G;n)$

2^{\ell_{2}}\ell_{2}!

of the

isomorphism classes

\cdots n^{\ell_{n}}\ell_{n}!)^{\beta(G)-1}$

.

of


107

For convenience, let denote the set of all partitions of the natural number . For a partition of , let denote the multiplicity of in the partition , so that . By using this terminology, Kwak and Lee [23] enumerate the number of the isomorphism classes of connected n-fold coverings as follows. (al so see Hofmeister [13]). $\mathfrak{P}(n)$

$n$

$\mathfrak{p}$

$n$

$k$

$j_{k}(\mathfrak{p})$

$j_{1}(\mathfrak{p})+2j_{2}(\mathfrak{p})+\cdots+nj_{n}(\mathfrak{p})=n$

$\mathfrak{p}$

THEOREM 9. ([23]) For $n\geq 2$ , the number of the isomorphism classes of connected n-fold covenngs of $G$ is Isoc $(G;n)$ $=\sum_{l_{1}+2l_{2}\cdots+(n-1)\ell_{*-1}=n-1}((t_{1}+1)^{\beta(G)-1}-1)$

$\times(f_{1} !

2^{\ell_{2}}\ell_{2}!\cdots(n-1)^{l_{n-1}}\ell_{n-1}!)^{\beta(G)-1}$

$+\sum_{2\ell_{2}+u_{\theta}+\cdots+n\ell.=n}(2^{\ell_{2}}\ell_{2}!3^{l_{\theta}}\ell_{3}!\cdots n^{\ell_{n}}\ell_{n}!)^{\beta(G)-1}$

$-\sum_{\mathfrak{p}\in \mathfrak{P}(n)-\{[[n,\cdot 1]]\}j_{k}}\prod_{(\mathfrak{p})\neq 0}$

(

$\frac{1}{j_{k}(\mathfrak{p})!}\prod_{\ell=0}^{j_{k}(\mathfrak{p})-1}$

(Isoc $(G;k)+\ell$

))

,

$j_{1}(p)=0$

where the summation over the empty index set is

defined to

be . $0$

Notice that the number of the isomorphism classes of connected n-fold coverings of $G$ is equal to the total number of the conjugacy classes of subgroups of index of the free group generated by elements. For any natural number , the enumeration of the isomorphism classes of regular n-fold coverings is much harder than that of all coverings. It has been done for only few cases. $n$

$\beta(G)$

$n$

THEOREM 10. ([9], [20], [32]) (a) The number of the isomorphism classes nected double coverings) of $G$ is $Iso^{R}(G;2)=2^{\beta(G)}$

and

of double covenngs

(resp. ISOC

(resp. con-

$(G;2)=2^{\beta(G)}-1$ ).

(b) For any pnme number , the number of the isomorphism classes of regular p-fold covemngs (resp. connected regular p-fold covenngs) of $G$ is $p$

$Iso^{R}(G;p)=\frac{p^{\beta(G)}+p-2}{p-1}$

and

.

$(resp$ $Isoc^{R}(G;p)=\frac{l^{(G)}-1}{p-1})$

.

J. LEE

108

The number of the isomorphism classes of A-coverings was computed when , etc. Almost all of these result ([20, 32]), $A=h^{2}$ ([21]), can be contained in the following.

$s$

$\oplus_{1=1}^{n}\mathbb{Z}_{p}([11])$

$\mathcal{A}=h$

THEOREM 11. ([18]) Let . . . $