Distinguishing partitions of complete multipartite graphs

0.27 MB·English

by  Michael Goff

DISTINGUISHING PARTITIONS OF COMPLETE
MULTIPARTITE GRAPHS
3
1 MICHAELGOFF
0
2
Abstract. AdistinguishingpartitionofagroupXwithautomorphismgroup

n aut(X) is a partition of X that is fixed by no nontrivial element of aut(X).
a In the event that X is a complete multipartite graph with its automorphism
J group,theexistenceofadistinguishingpartitionisequivalenttotheexistence

9 ofanasymmetrichypergraphwithprescribededgesizes. Anasymptoticresult
1 isprovenonthe existence ofadistinguishingpartitionwhenX isacomplete
multipartitegraphwithm1 partsofsizen1 andm2 partsofsizen2 forsmall
] n1, m2 and largem1, n2. A key tool inmaking the estimate is counting the
O
numberoftreesofparticularclasses.
C
.
h
t
a 1. Introduction
m
The distinguishing partition problem asks, given a finite set X with a group
[
aut(X) that acts on X, whether there exists a partition P of the elements of X

1 such that no nontrivial element of aut(X) fixes P. Formally, consider a parti-
v
tion P = P ,...,P and γ aut(X). For general X′ = x ,...,x X, let
3 { 1 t} ∈ { 1 i} ⊂
γ(X′)= γ(x ),...,γ(x ) . Then let γ(P)= γ(P ),...,γ(P ) . We say that P is
8 1 i 1 t
{ } { }
5 a distinguishing partition if γ(P)=P for all nontrivial γ aut(X).
6 ∈
4 Not all sets X with group action aut(X) have a distinguishing partition. For
1. example,if aut(X)is the groupof allpermutation onX and X 2, then X does
| |≥
0 nothavea distinguishing partition. Conversely,if aut(X) is the trivialgroup,then
3 all partitions of X are distinguishing. In general, the conditions for the existence
1
of a distinguishing partition can be quite complex, even in a relatively restricted
:
v setting such as taking X to be a complete multipartite graph, acted upon by its
i automorphism group. Informally, the difficulty is that if a partition P consists of
X
few large parts, then a nontrivial automorphism might fix each part, while if P
r
a consists of many small parts, then a nontrivial automorphism might permute the
parts.
Ellingham and Schroeder [7] first considered the distinguishing partitions prob-
lem for complete equipartite graphs. Their finding is that if X is a complete
equipartite graph with m parts, each of size n, then X has a distinguishing par-
tition if and only if m f(n) for f(2) = f(14) = 6, f(6) = 5, and otherwise
≥
f(n) = log (n+1) +2. In this setting, aut(X) is the wreath product action
⌊ 2 ⌋
S Wr S .
n m
The distinguishing partition is a measure of the level of symmetry of a group
action, and as such the concept is closely related to the well-studied distinguishing
number, as introduced by Albertson and Collins [1] on a graph and by Tymoczko
[11]forageneralgroupaction. Othersuchmeasuresarethecostof2-distinguishing
Date:November25,2012.
1
2 MICHAELGOFF
[6] and the determining set [5]. The survey of Bailey and Cameron [2] shows how
these concepts have appeared independently in many different settings.
For the remainder of this paper, we will consider the case that X is a complete
multipartite graph with its automorphism group. We denote by X = K
n1,...,nm
the complete multipartite graph with maximal independent sets X of size n for
i i
1 i m. Also, K denotes the complete multipartite graph with m
≤ ≤ m1(n1),m2(n2) i
parts eachof size n for i=1,2. We focus in particularon K for fixed
i m1(n1),m2(n2)
n and m and large m and n .
1 2 1 2
BasedontheresultsofEllinghamandSchroeder[7],wemightexpectacomplete
multipartite graph to have a distinguishing partition if it has many small parts,
and not to have a distinguishing partition if it has few large parts. In our setting,
whichcombinesthesetwoextremes,itseemsnaturaltoexpectthatadistinguishing
partition, in the asymptotic sense, would exist if n /m does not exceed a certain
2 1
ratio. Our main result is that this is indeed the case.
Theorem 1.1. Fix n 2 and m 1, and suppose that m is large. There exists
1 2 1
≥ ≥
avaluer =r suchthatthefollowingholds. K hasadistinguishing
n1,m2 m1(n1),m2(n2)
partition if and only if
n r m +ǫ(m )
2 ≤ n1,m2 1 1
for some function ǫ(m ) o(m ).
1 1
∈
We have that r =1. For n 3, we define r by first choosing values of
2,m2 1 ≥ n1,m2
j =j and k =k such that
n1,m2 n1,m2
m m m
2 2 2
n =2+ + + + +k,
1
0 1 ··· j
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
with either
m
2
j < (m 1)/2 and 0 k < , or j = (m 1)/2 and k 0.
2 2
⌊ − ⌋ ≤ j+1 ⌊ − ⌋ ≥
(cid:18) (cid:19)
If j < (m 1)/2 , then let
2
⌊ − ⌋
j
m i m m j 1
2 2 2
r =1+ − + − − k,
m i m
i=0 2 (cid:18) (cid:19) 2
X
and otherwise choose
j
m i m 1
2 2
r =1+ − + k.
m i 2
i=0 2 (cid:18) (cid:19)
X
We say that j = 1.
2,m2 −
The structure of the paper and the proof Theorem 1.1 is as follows. In Section
2, we establish basic concepts on enriched trees and hypergraphs which are used
heavily throughout the proof. In Section 3, we show how a type of partition of
K known as a regular partition may be represented as a hypergraph
m1(n1),m2(n2)
with m edges of size n , i = 1,2. We establish key lemmas for the general result
i i
in Section 4. In Section 5, we provide the general construction that maximizes n
2
to within an additive constant, given m ,n ,m . Then we prove that for large m
1 1 2 1
relativeton andm ,ifn′ >n andK hasadistinguishingpartition,
1 2 2 2 m1(n1),m2(n′2)
then so does K .
m1(n1),m2(n2)
In Section 6, we focus on the case that n = 2. Then the following refinement
1
of Theorem 1.1 holds.
DISTINGUISHING PARTITIONS OF COMPLETE MULTIPARTITE GRAPHS 3
Theorem 1.2. There exist constants α>0 and β >1 and
m1(β 1) 3/2
z := log − log m
β αβ β 1
(cid:22) (cid:18) (cid:19)(cid:23)
such that Theorem 1.1 holds with ǫ(m ) of th(cid:0)e form (cid:1)
1
2
m β m
1 +(1+o (1))αβzz−7/2 1 .
z+1 m1 β 1 ≈ log (m )
(cid:18) − (cid:19) β 1
In Section 7, we consider the case that k = 0 and j < (m 1)/2 . Then
2
⌊ − ⌋
Theorem 1.1 can be refined as follows.
Theorem 1.3. If k =0 and j < (m 1)/2 , then Theorem 1.1 holds with ǫ(m )
2 1
⌊ − ⌋
of the form
(2m22−m4j−44j)22mm522−−44jj−−54C2m2−14j−4 +om1(1) m122mm22−−44jj−−45
2 !
− −
for a value of C that depends only on n and m .
1 2
The value of C will be specified in Section 7.
We consider k 1 and j < (m 1)/2 in Section 8.
2
≥ ⌊ − ⌋
Theorem 1.4. If k 1 and j < (m 1)/2 , then Theorem 1.1 holds with ǫ(m )
2 1
≥ ⌊ − ⌋
of the form Θ(m /(logm )).
1 1
In Section 9, we consider the case that k = 0 and j = (m 2)/2. Then the
2
−
following exact result for large m is possible.
1
Theorem 1.5. Suppose that k =0 and j = (m 1)/2 . Theorem 1.1 holds with
2
⌊ − ⌋
ǫ(m1)=2m2−1 if m2 is even and at least 4 and ǫ(m1)=2m2−1 1 if m2 is odd or
−
2 for sufficiently large m .
1
In Section 10, we prove the following for k 1 and j = (m 1)/2 .
2
≥ ⌊ − ⌋
Theorem1.6. Ifk 1andj = (m 1)/2 ,thenTheorem1.1holdswithǫ(m )=
2 1
≥ ⌊ − ⌋
2m2−1 1ifkm1 isevenandm2 isodd,andotherwiseǫ(m1)=2m2−1+ rm1 rm1,
− ⌊ ⌋−
for sufficiently large m .
1
2. Enriched trees and hypergraphs
Combinatorial species and enriched trees. We make use of the language of
combinatorialspecies,aspresentedbyBergeron,Labelle,andLeroux[3]. Aspecies
F is, for every finite set U, a finite set of objects F[U], called structures, together
with, for every bijection σ : U U′, a function F[σ]: F[U] F[U′] that satisfies
→ →
the following two functoriality properties:
1) for all bijections σ :U U′ and σ′ :U′ U′′, F[σ′ σ]=F[σ′] F[σ],
→ → ◦ ◦
2) for the identity map Id , F[Id ]=Id .
U U F[U]
The function F[σ] is known as transport of species. Consider the symmetric group
S that acts on U. Given an F-structure s, we say that the automorphism group
U
of s, aut(s), is the subgroup of those σ S that satisfy F[σ](s)=s.
U
Let a be the species of asymmetric ∈trees, or trees whose automorphism group
is trivial, and let a· be the species of rooted asymmetric trees. A rooted tree is
considered asymmetric if it has no nontrivial root-preserving automorphism; it is
possible that the underlying unrooted tree structure is not asymmetric.
4 MICHAELGOFF
Now let F be a species that contains at least one structure over a set of size 1.
AnF-enriched tree onasetU isatreeonU togetherwithanF-structures onthe
v
neighborsetN(v)ofeveryvertexv U. Ifσ isanautomorphismofanF-enriched
∈
treet,thenσisanautomorphismoftheunderlyingtreestructureoft. Furthermore,
if σ is the restrictionofσ onN(v), then the transportof species F[σ ]takes s to
v v v
s . We say that a is the species of asymmetric F-enriched trees. For example,
σ(v) F
when F is , the species of sets, then there is a unique -structure on every finite
set, and a Eis simply a. The species A and A are,respeEctively, the species of (not
E F
necessarily asymmetric) trees and the species of F-enriched trees.
The sum oftwo species (F +F′)[U] is the disjoint unionF[U]+F′[U] suchthat
(F +F′)[σ](s)=F[σ](s) if s F[U] and (F +F′)[σ](s)=F′[σ](s) if s F′[U]. If
∈ ∈
a is a positive integer, then aF is the sum of a copies of F. We say that F [U] is
i
F[U] when U = i, and otherwise F [U] = , and F is the number of structures
i i
| | ∅ | |
of F over a set with i elements.
Consider the species F = κ a for nonnegative integers κ,a ,...,a with
i=1 iEi 1 κ
κ 1 and a 1. This is the species that consists of a distinct set structures
1 i
≥ ≥ P
over a set with i elements for 1 i κ and otherwise no structures. Then the
species a may be regarded as ≤the ≤species of asymmetric trees in which every
F
vertex has degree at most κ, and every vertex of degree i is assigned a label from
a pool of a possible labels. Such a tree is asymmetric if it has no nontrivial label-
i
preserving automorphism. Later, we show how estimating the number of elements
of a with a given number of vertices can help determine asymptotic bounds for
F
the distinguishing partitions problem.
A structure of the product species FF′ over U is an ordered pair (f,f′) for an
F-structure f over U and a F′-structure f′ over U for some partition U U of
1 2 1 2
⊔
U. Transport of species is defined by (FF′)[σ](s)=(F[σ ](f),F′[σ ](f′)), each σ
1 2 i
the restriction of σ to U .
i
Hypergraphs. Ahypergraph H isatriple(V(H),E(H),I(H)),withV(H)afinite
set of elements called vertices and E(H) a finite set of elements called edges. The
incidence relation I(H) is a subset of V(H) E(H). We will generally treatedges
×
as subsets of V(H). The degree of v V(H), denoted deg(v), is the number of
∈
edges incident to v. H is connected if the bipartite graph with vertex sets V(H)
andE(H)andedgesetI(H)isconnected,and E(H) = . Ifeveryedgeisincident
| |6 ∅
to n -vertices, then H is n -uniform.
1 1
For a hypergraph H with an edge e, deg (H) and deg (e) denote the number
1 1
of vertices of degree 1 in H and e, while deg+H and deg+e are the numbers of
2 2
vertices of degree at least 2 in H and e.
Anautomorphismσ ofH isapermutationofV(H)andE(H)suchthat(v,e)
∈
I(H) if and only if (σ(v),σ(e)) I(H) for all vertices v and edges e. We say
∈
that σ is trivial if σ(v) = v and σ(e) = e for all vertices v and edges e, and
H is asymmetric if the only automorphism of H is trivial. Thus we allow that
hypergraphs may contain multiple edges that are incident to the same vertex set,
but such hypergraphs are not asymmetric.
A connected n -uniform hypergraph H is a tree if E(H) can be enumerated
1
(e ,...,e ) in such a way that for each 2 i E(H), we have that e
1 |E(H)| i
≤ ≤ | | | ∩
(e e ) = 1. Equivalently, a tree is a connected n -uniform hypergraph
1 i−1 1
∪···∪ |
H with (n 1)E(H) +1 vertices. The leaves of a tree H are the edges e that
1
− | |
satisfy deg (e)=n 1. We say that l(G) is the number of leaves of G.
1 1−
DISTINGUISHING PARTITIONS OF COMPLETE MULTIPARTITE GRAPHS 5
For a tree G, define the quantity
µ(G):= (deg(v) 2).
−
v∈XV(G)
deg(v)>2
We will later need the following relationship between l(G) and µ(G).
Lemma 2.1. Let G be a tree with at least 2 edges. Then l(G) µ(G)+2.
≥
Proof: Enumerate E(G)=(e ,...,e ) such that for 2 i E(G),
1 |E(G)|
≤ ≤| |
e (e ... e )= v ,
i 1 i−1 i
∩ ∪ ∪ { }
and let G be the subtree of G with edges (e ,...,e ). We prove that the lemma
i 1 i
holds for G by induction on i for 2 i E(G), with the i = 2 case following
i
≤ ≤ | |
from µ(G )=0 and l(G )=2. Assume the lemma holds for G .
2 2 i−1
For 3 i E(G), µ(G ) = µ(G )+1 if v has degree at least 3 in G ,
i i−1 i i
≤ ≤ | |
and otherwise µ(G ) = µ(G ). Also, l(G ) l(G ) since e is a leaf in G ,
i i−1 i i−1 i i
≥
and at most one leaf of G , namely a leaf that contains v as a degree 1 vertex,
i−i i
is not a leaf in G . Furthermore, whenever µ(H ) = µ(G )+1, v has degree
i i i−1 i
at least 2 in G and thus l(G ) = l(G )+1. The lemma follows for G by
i−1 i i−1 i
l(G ) l(G ) µ(G ) µ(G ). (cid:3)
i i−1 i i−1
− ≥ −
3. Distinguishing partitions and asymmetric hypergraphs
We demonstrateabijectionbetweendistinguishingpartitionsofcompletemulti-
partite graphs and asymmetric hypergraphs with prescribed edge sizes. Using this
bijection, we establish the existence or nonexistence of distinguishing partitions by
demonstrating the existence or nonexistence of certain asymmetric hypergraphs.
The argument is nearly identical to that given by Ellingham and Schroeder [7].
WithX =K ,letP beapartitionofX withpartsP ,...,P ,andwesay
n1,...,nm 1 t
that P is a regular partition of X if X P 1 for all i and i′. It is a necessary
i i′
| ∩ |≤
but not sufficient condition for P to be distinguishing that P is regular.
Definition 3.1. For every regular partition P of X with parts P ,...,P , we as-
1 t
sociate a hypergraph τ(P) as follows: V(τ(P)) = P ,1 i t , E(τ(P)) =
i
{ ≤ ≤ }
X :1 i m , and X and P are incident if X P =1.
i i i′ i i′
{ ≤ ≤ } | ∩ |
Note that τ(P) is a hypergraph with m edges with sizes n ,...,n since X
1 m i
intersects exactly n parts of P.
i
Wesaythattheautomorphismgroupaut(P)isthesubgroupofaut(X)consisting
of those elements that fix P. The following relationship holds.
Lemma 3.2. If P is a regular partition of X, then aut(P) is isomorphic to
aut(τ(P)).
Proof: Let τ˜ : aut(P) aut(τ(P) be the group homomorphism induced by τ.
→
Say that P ,...,P are the parts of P.
1 t
An automorphism σ aut(P) induces automorphisms σ and σ on the sets
P X
∈
P and X respectively. Then σ is uniquely determined by σ and σ , and in
i i′ P X
{ } { }
particularσ is trivialif andonly if σ and σ areboth trivial. Thus τ˜ is injective.
P X
Nowletσ′ aut(τ(P)). Thenσ′ isuniquelydeterminedbyincidence-preserving
∈
permutationsof P and X . Letσ bethe permutationofX suchthatifx X
i i′
{ } { } ∈
isthe unique vertexcontainedinX P ,thenσ(x) is the unique vertexcontained
i i′
∩
6 MICHAELGOFF
in σ′(X ) σ′(P ). It is readily checked that in fact σ aut(P), and thus τ˜ is
i i′
surjective.∩ ∈ (cid:3)
Corollary 3.3. There exists a distinguishing partition of K if and only if
n1,...,nm
there exists an asymmetric hypergraph with m edges of sizes n ,...,n .
1 m
It will be convenient to associate another hypergraph with a regular partition
P of K . Let τ′(P) be a vertex-labeled hypergraph that contains ex-
m1(n1),m2(n2)
actly the vertices and the n -edges of τ(P). Say that the n -edges of τ(P) are
1 2
X1,...,Xm2. Then the vertex label set of τ′(P) is 2[m2], and a vertex v in τ′(P)
is labelled with a set S [m ] if v X exactly when i S. Then τ′(P) is just a
2 i
⊆ ∈ ∈
different way of encoding τ(P).
The weight w(v) of a vertex v τ′(P) is the cardinality of its label. The weight
∈
w(S)ofasetofverticesS isthesumoftheweightsoftheverticesinS. Theweight
w(e) or w(G) of an n -edge e or connected component G τ′(P) is the weight of
1
⊂
thevertexsetofeorG. Thevalue ofGisw(G) rm E(G). Valuemaybepositive
2
− | |
ornegative. Wehavethatn =w(τ′(P))/m ,andthusourstrategyinprovingthe
2 2
main results is to find an asymmetric labelled hypergraph with m n -edges and
1 1
maximal weight. Though weight and value encode the same information, value is
useful in that it gives a clear comparisionof the weight of a component of τ′(P) to
its asymptotic limit.
4. Key Lemmas
We now present a series of lemmas that provide upper bounds on the weights
and values of certain types of components.
Lemma 4.1. Let G be a connected n -uniform hypergraph with n E(G)/2+p
1 1
| |
vertices. Then G contains 2p+µ(G) vertices of degree 1. Equivalently, each edge
has, on average, (2p+µ(G))/E(G) degree 1 vertices.
| |
Proof: Therearen E(G) pairsofthe form(v,e), wherev is avertex, eanedge,
1
| |
and v e. The number of such pairs (v,e) is also
∈
deg (G)+2(n E(G)/2+p deg (G))+µ(G),
1 1| | − 1
or n E(G) +2p deg (G)+µ(G). Thus deg (G)=2p+µ(G). (cid:3)
1| | − 1 1
Lemma 4.2. Suppose that τ(P) is asymmetric, and let S be the set of degree 1
vertices in an edge of τ′(P). Then S 2m2. Define nonnegative values j′ and k′
such that S = m2 + + m2 +| k|′≤with either 0 k′ < m2 or k′ = 0 and
| | 0 ··· j′ ≤ j′+1
j′ =m . Then
2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
j′
m
w(S) (m i) 2 +k′(m j′ 1).
2 2
≤ − i − −
i=0 (cid:18) (cid:19)
X
Proof: For all v ,v S, there is an automorphism of the underlying unlabelled
1 2
∈
hypergraph of τ′(P) that switches v and v and fixes all other vertices. Thus all
1 2
verticesinS musthavedifferentlabelsinτ′(P),whichimpliesthat S 2m2. The
lemma follows from the fact that S contains at most m2 vertices|wi|t≤h a label of
i
cardinality m i. (cid:3)
2
− (cid:0) (cid:1)
DISTINGUISHING PARTITIONS OF COMPLETE MULTIPARTITE GRAPHS 7
Let w denote the upper bound on w(S) in Lemma 4.2. Now suppose that
|S|
τ′(P) is asymmetric and G is a component of τ′(P). A defect in G is one of the
following. A defective vertex is a vertex v with degree at least 2 and weight less
than m , counted with multiplicity d(v)=m w(v). A defective edge is an edge
2 2
−
e with setS ofdegree1 vertices with collectiveweightless than w , countedwith
|S|
multiplicity d(e)=w w(S). The number of defects in G is denoted by d(G).
|S|
−
Lemma 4.3. Let G be a connected component of τ′(P). If τ(P) is asymmetric,
then
w(G) m deg+(G)+ w d(G).
≤ 2 2 deg1(e)−
e∈XE(G)
Proof: Everyvertex has weight at most m , and a defective vertex v with degree
2
atleast2hasweightm d(v). Thesetofdegree1verticesinanedgeehasweight
2
−
w if e is not defective, and otherwise weight w d(e). The lemma
deg (e) deg (e)
follow1 s by adding over all vertices. 1 − (cid:3)
If G contains n E(G)/2+p vertices, then write
1
| |
2p+µ(G) 2p+µ(G)
2p+µ(G)=b +b′
E(G) E(G)
(cid:22) | | (cid:23) (cid:24) | | (cid:25)
with nonnegative b+b′ = E(G). The following lemma states that the weight of
| |
G is maximized when all edges have about the same number of degree 1 vertices.
Lemma 4.4. With all quantities as above,
w(G) m n E(G)/2 m p m µ(G)+bw +b′w d(G).
≤ 2 1| | − 2 − 2 2p+µ(G) 2p+µ(G) −
⌊ |E(G)| ⌋ ⌈ |E(G)| ⌉
Proof: By Lemma 4.1, G has n E(G)/2 p µ(G) verticesof degree atleast 2.
1
| | − −
Then by Lemma 4.3,
w(G) m n E(G)/2 m p m µ(G)+ w d(G).
2 1 2 2 deg (e)
≤ | | − − 1 −
e∈XE(G)
The expressionw is concaveiny, meaning that for all y, w w w w .
y y y−1 y+1 y
− ≥ −
Thus,givena set of values y suchthat y =2p+µ(G), t w is maximal
{ i} i i i=1 yi
whenb ofthe y areequalto 2p+µ(G) andb′ ofthe y are 2p+µ(G) . The lemma
i |E(G)| P i P|E(G)|
follows. j k l m (cid:3)
We now look to maximize the weight of G by considering the total number of
vertices of a given weight. In particular, G contains at most E(G) m2 degree
| | i
1 vertices with weight m i, since each each edge contains at most m2 such
2 − (cid:0) (cid:1)i
vertices. Choose values j∗ and k∗ such that
(cid:0) (cid:1)
m m
2p+µ(G)= E(G) 2 + + E(G) 2 +k∗
| | 0 ··· | | j∗
(cid:18) (cid:19) (cid:18) (cid:19)
with j∗ 1 and 0 k∗ < E(G) m2 .
≥− ≤ | | j∗+1
Lemma 4.5. With all quantities a(cid:0)s abo(cid:1)ve, w(G)
≤
j∗
n E(G) m
m 1| | p µ(G) + E(G)(m i) 2 +(m j∗ 1)k∗ d(G).
2 2 2
2 − − | | − i − − −
(cid:18) (cid:19) i=0 (cid:18) (cid:19)
X
8 MICHAELGOFF
Proof: Let G′ be a (not necessarily asymmetric) hypergraphconstructed from G
by giving every vertex of degree at least 2 the label [m ] and assigning a label to
2
every degree 1 vertex such that all edges of G′ are nondefective and have distinct
labels among the degree 1 vertices. Then G′ has (n E(G)/2 p µ(G)) vertices
1
ofdegreeatleast2,eachofwhichhasweightm , and| atm|ost−E(G−) m2 degree1
2 | | i
verticesofweightm ifor0 i j∗. Theresultfollowsbyw(G′)=w(G)+d(G).
2
(cid:3) − ≤ ≤ (cid:0) (cid:1)
We consider the upper bound of Lemma 4.5 to be a function w (p,µ(G),d),
max
with the quantities m and E(G) considered to be fixed. We note that
2
| |
w (p,µ(G),d)>w (p,µ(G),d+1),
max max
while
w (p,µ(G),d) w (p,µ(G)+1,d),
max max
≥
with equality exactly when j∗ = 1.
−
Nowweconsiderµ(G)=d=0,andtheupperboundofLemma4.5isafunction
w (p). The effect of replacing p by p+1 is equivalent, numerically, to replacing
max
a vertex with weight m by two vertices, one of weight m j∗ 1 and the other
2 2
− −
of weight either m j∗ 1 or m j∗ 2. Thus the function w (p) is weakly
2 2 max
− − − −
unimodal in p and achieves a maximum when j∗ = (m 1)/2 and k∗ =0 or 1.
2
⌊ − ⌋
Then 2p= ⌊(m2−1)/2⌋ E(G) m2 +(0 or 1), and we have by Lemma 4.5 that
i=0 | | i
P (cid:0) (cid:1)⌊m2−1⌋ ⌊m2−1⌋
m n m 2 m 2 m
2 1 2 2 2
(1) w(G) E(G) + (m i) .
2
≤| | 2 − 2 i − i 
i=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19)
X X
Thus the following holds. 
Corollary 4.6. Let all quantities be as above.
(1) If j = (m 1)/2 , then w(G) m rE(G).
2 2
(2) If j <⌊(m −1)/2⌋ and p E(≤G)(n1| 1),|then w(G) m rE(G).
⌊ 2− ⌋ ≤| | 2 − ≤ 2 | |
(3) G has positive value only if j < (m 1)/2 and G is a tree. Then if G
2
⌊ − ⌋
has d defects, v(G) m 2j d.
2
≤ − −
Proof: Part1 followsbyEquation(1) andthe definition ofr. Part2 followsfrom
the monotonicity of w and the definition of r. Part 3 is a consequence of Parts
max
1 and 2. (cid:3)
We now focus on the particular case that G is a tree and k = 0. Suppose that
G has l leaves, and by Lemma 2.1, l µ(G)+2. Then j∗ =j and k∗ =2+µ(G),
≥
and w
e∈E(G) deg1(e) ≤
P j m
2
E(G)(m i) +(µ(G)+2)(m j 1).
2 2
| | − i − −
i=0 (cid:18) (cid:19)
X
This bound can be attained if G contains E(G) m2 degree 1 vertices of weight
| | i
m i for 0 i j andµ(G)+2 degree 1 vertices of weightm j 1. However,
2 2
− ≤ ≤ (cid:0) (cid:1) − −
every leaf of G contains a vertex of weight at most m j 1, and thus in fact
2
− −
w
e∈E(G) deg1(e) ≤
P j m
2
E(G)(m i) +(µ(G)+2)(m j 1) (l µ(G) 2).
2 2
| | − i − − − − −
i=0 (cid:18) (cid:19)
X
DISTINGUISHING PARTITIONS OF COMPLETE MULTIPARTITE GRAPHS 9
Since G has E(G) 1 µ(G) vertices of degree 2 or more,
| |− −
j
m
2
w(G) m E(G) + E(G)(m i) µ(G)j+m 2j l.
2 2 2
≤ | | | | − i − − −
i=0 (cid:18) (cid:19)
X
Finally, ifwe allowthatG mighthaveddefects, then weconclude the following.
Lemma 4.7. With all quantities as above, v(G) m 2j l jµ(G) d.
2
≤ − − − −
Now weconsiderallofτ′(P). LetComp(P)be the setofconnectedcomponents
of τ′(P).
Lemma 4.8. If τ(P) is asymmetric, v(τ′(P))≤ G∈Comp(P)v(G)+m22m2−1.
Proof: We calculate that v(τ′(P)) is P v(G) plus the sum of the
G∈Comp(P)
weights of all vertices not contained in any n -edge. Since τ(P) is asymmetric,
P 1
every vertex not contained in an n -edge must have a different label, and there is
1
at most one vertex with every label S 2[m2]. The lemma follows. (cid:3)
⊂
5. An extremal construction
In this section, we give a general method of constructing a distinguishing parti-
tion P of K . We then show that n is maximal to within an additive
m1(n1),m2(n2) 2
constant,giventheotherparameters. Wedosobydescribingthevertex-labeledhy-
pergraphτ′(P). Fortheremainderofthissection,weassumethatj < (m 1)/2 ;
2
⌊ − ⌋
the case that j = (m 1)/2 is treated seperately.
2
⌊ − ⌋
LetGbeacomponentofτ′(P). Define thevalue v(e)ofanedgee E(G)tobe
∈
v(G)/E(G). Let ξ be the map that adds 1 mod m to every element in the label
2
| |
of every vertex of a 2[m2]-vertex lableled hypergraph. Note that v(ξ(G)) = v(G).
Say that vertex labeled hypergraphs G,G′ are equivalent under if G = ξi(G′)
ξ
∼
for some i.
Let ∗ = ∗ = ( , ,...) be an ordered list of equivalence classes under
T Tn1,m2 T1 T2
of positive weight asymmetric hypergraphs such that an edge in an element of
ξ
∼
has value at least as great as an edge in an element of for all i. By Lemma
i i+1
T T
4.7, an edge e may have value δ > 0 only if e is contained in a tree with at most
m /δ edges. Thus ∗ enumerates all hypergraphs with positive value, and only
2
T
classes of trees are in ∗.
T
A symmetry breaking ring R is a 2[m2]-vertex labeled hypergraph on at least
min = min (n ,m ) edges defined as follows. If m = 1, then j = 1 and
R R 1 2 2
−
n = 2, and we set min = 6. Let R be a cycle that contains consecutive vertices
1 R
v ,v ,v ,v . Then all vertices of R have weight 1 except for v ,v ,v , which all
1 2 3 4 1 2 4
have weight 0.
If m > 1, we set min = max(2m ,n + 1). Let quot be the maximum
2 R 2 1 R
multipleofm2 upto|E(R)|. Letv0,...,vquotR−1 beverticesande0,...,equotR−1
be edges such that e contains only degree 1 vertices except for v ,v , subscripts
i i i+1
mod quot . The set of vertex labels of the degree 1 vertices of e includes all
R 0
possible labels of size at least m j, and all others are of size m j 1. To
2 2
− − −
determine the labels of the degree 1 vertices of e , add i mod m to every element
i 2
inthelabelsofthedegree1verticesofe . Assignv thelabel ,v thelabel[m ] i
0 0 i 2
∅ −
for 1 i m , and v the label [m ] for all other i. Finally, R contains edges of
2 i 2
≤ ≤
the form v ,v ,...,v for 1 i E(R) quot .
i i+1 i+n1−1 ≤ ≤| |− R
We need some key facts on symmetry breaking rings.
10 MICHAELGOFF
Lemma 5.1. Let R be a symmetry breaking ring that is a component of τ′(P).
Then no automorphism of P induces a nontrivial automorphism of R.
Proof: Thelemmaisreadilyverifiedwhenm =1,andsoweassumethatm >1.
2 2
Let σ be an automorphism of τ(P) that induces an automorphism of R. Since v
0
is the only vertexof R ofweight0,it is a fixedpoint. Since v is the only vertex of
1
R that is in a common edge with v , has degree 2 in τ′(P), and weight not equal
0
to m , v is also a fixed point. Thus all v are fixed, which implies that σ fixes the
2 1 i
n -edges of τ(P). Since all v are fixed, σ thus also fixes all n -edges of R. Finally,
2 i 1
since all degree 1 vertices in a given edge have different labels, they must be fixed
points as well. (cid:3)
The following is readily observed from the construction of symmetry breaking
rings.
Lemma 5.2. Let R be a symmetry breaking ring, and let 1 i < i′ m . Then
2
≤ ≤
the number of vertices of R whose label contains i is equal to the number of vertices
of R whose label contains i′.
Lemma 5.3. There exists a value ω =ω , which depends only on n and m ,
n1,m2 1 2
such that a symmetry breaking ring R has value at least ω.
Proof: If m =1, then v(R)= 3. If m >1, then the totalweightof the degree
2 2
−
1 vertices in each edge with degree 1 vertices is m (r 1). All other vertices have
2
−
weightm exceptform verticesofweightm 1andoneofweight0. Itfollowsthat
2 2 2
−
w(R)=quot m r 2m andv(R)= (E(R) quot )m r 2m > m2r 2m .
(cid:3) R 2 − 2 − | |− R 2 − 2 − 2 − 2
WenowcometoourconstructionofP. Chooseζ tobethemaximumvaluesuch
that the total number of edges in all trees of is at most m min .
1 ζ 1 R
T ∪···∪T −
Let ∆ be the union of all trees in , together with a symmetry
m1(n1),m2 T1∪···∪Tζ
breakingringsothat∆ hasm edges,andadegree0vertexofeverylabel
m1(n1),m2 1
except .
∅
Lemma 5.4. ∆ is in fact τ′(P) for a distinguishing partition P for an
m1(n1),m2
appropriate value of n .
2
Proof: By construction, ∆ contains the same number of vertices whose
m1(n1),m2
labelcontainsiasthenumberofverticeswhoselabelcontainsi′ forall1 i<i′
≤ ≤
m . Thus ∆ is τ′(P) for some partitionP of K and for some
2 m1(n1),m2 m1(n1),m2(n2)
n . Next, we apply Corollary 3.3 and show that P is distinguishing by showing
2
that τ(P) is asymmetric. Let σ be an automorphism of τ(P). Since ∆
m1(n1),m2
contains exactly one symmetry breaking ring, all n -edges of τ(P) are fixed under
2
σ. Since all components of ∆ are asymmetric and no two are isomorphic
m1(n1),m2
to each other, all n -edges and vertices of τ(P) are fixed as well. (cid:3)
1
Next, we prove that ∆ is a nearly optimal construction.
m1(n1),m2
Lemma5.5. LetP beadistinguishingpartitionofK suchthatτ′(P)=
m1(n1),m2(n2)
∆ , and let G′ be an asymmetric vertex-labeleld hypergraph. Then w(G′)
m1(n1),m2 ≤
w(∆ ) + Error for some value Error that depends only on n
m1(n1),m2 n1,m2 n1,m2 1
and m . In particular, if P′ is a distinguishing partition of K , then
2 m1(n1),m2(n′2)
n′ n +Error /m .
2 ≤ 2 n1,m2 2