List Multicolourings of Graphs with Measurable Sets
A.J.W. Hilton
Department of Mathematics
The University of Reading
Whiteknights, Reading RG6 6AX
England
P.D. Johnson, Jr.
Department of Discrete
and Statistical Sciences
Auburn University, Alabama 36849
USA
Abstract
In an ordinary list multicolouring of a graph, the vertices are “coloured”
with subsets of pre-assigned finite sets (called “lists”) in such a way that adjacent
vertices are coloured with disjoint sets. Here we consider the analogue of such
colourings in which the lists are measurable sets from an arbitrary atomless,
semifinite measure space, and the colour sets are to have prescribed measures rather
than prescribed cardinalities. We adapt a proof technique of Bollobás and
Varopoulos to prove an analogue of one of the major theorems about ordinary list
multicolourings, a generalization and extension of Hall’s marriage theorem, due to
Cropper, Gyárfás, and Lehel.
1.
Introduction: List Multicolourings of Graphs
Throughout, G will be a finite, simple graph with vertex set V. When another graph
H enters the discussion, its vertex set will be denoted V(H). For other notation usual in
graph theory, see [10].
In the usual sense, list (multi) colouring problems arise in the following context. Let
C be an infinite set (of “colours”, or “symbols”), and let F denote the collection of finite
subsets of C. A function L : V  F is called a list assignment to the vertices of G: there
is also a function  : V  Ã  {1, 2, ...} , the cardinality prescription or colour demand
function. Given these, a proper ( L,  ) -colouring (or multicolouring; some reserve
“colouring” for the case   1 ) of G is a function  :V  F
satisfying
2
(i)
 (v)  L(v) for each v V ;
(ii)
 (v)   (v) for each v V ; and
(iii)
if u, v V are adjacent in G, then  (v)   (u )  0/ .
It is useful to note that (iii) is equivalent to:
(iii)
for each   C , supp ( , G)  v V ;   (v) is an independent set of vertices
of G. (That is, no two vertices of this set are adjacent in G .)
For   C and H a subgraph of G, let  ( , L, H ) denote the vertex
independence number of  supp L ( , H )  H , the subgraph of H induced by those vertices
bearing  on their L lists. Note that  ( , L, H ) is an upper bound on the number of
appearances of  in any proper (L, )-colouring of H. This interpretation leads easily to
the conclusion that the following, Hall’s condition on (G, L,  ) , is necessary for the
existence of a proper (L, )-colouring of G:
For each subgraph H of G,

vV ( H )
 (v ) 
  ( , L, H )
* (L, , H)
 C
It is useful to note that Hall’s condition is satisfied if * (L, , H) holds for each
induced subgraph H of G. The reason for the name “Hall’s condition” is that Hall’s
marriage theorem [5] may be stated: if G is a clique and   1 , then Hall’s condition, a
condition on L, is sufficient for the existence of a proper (L, 1)-colouring of G – which, in
these circumstances, is a “system of distinct representatives” of the sets L(v) , v  V .
A well-known strengthening of Hall’s theorem noted by many, among them Rado [9]
and Halmos and Vaughan [6], may be stated: if G is a clique, then Hall’s condition, now a
condition on L and , is sufficient for the existence of an (L, )-colouring of G. The
problem of characterizing the graphs which have this property of cliques has recently been
solved by Cropper, Gyárfás and Lehel [3]. The characterization is given in the following
theorem.
Theorem CGL ([3]; also, see [2].) The following are equivalent:
3
(a)
G has the property that Hall’s condition (on L and  ) is sufficient for the existence
of a proper (L,  )-colouring of G ;
(b)
G is the line graph of a forest;
(c)
every block of G is a clique, and every cut-vertex of G is in exactly two blocks;
(d)
G has none of Ck , k  4, K 4 -minus-an-edge, nor K1,3 as an induced subgraph.
In the next section we generalize list multicolouring problems, and Hall’s condition,
by allowing the “lists” to be measurable sets from any positive measure space, with the
colour sets to be measurable subsets of the lists of prescribed measure. In Section 3 we adapt
a proof of Bollobás and Varopoulos to prove the analogue of Theorem CGL when the
measure space is atomless and semifinite.
2.
List Multicolourings with Measurable Sets
Throughout, (X, , ) will be a positive measure space. In this notation, X is a set,  is a
-algebra of “measurable” subsets of X, and  : p  [0, ] is a positive measure. In
some places, in what follows, mention of  will be suppressed. If “  (Y ) ” appears, it
should be understood that “ Y p ” is implied.
Given a “list assignment” L : V  p
and a measure prescription  : V  [0, ) , a
proper (L, )-colouring of G is a function  :V  p
satisfying
(i)
 (v)  L(v) for each v V ;
(ii)
  (v)    (v) for each v V ; and
(iii)
if u, v V are adjacent in G, then  (v)   (u )  0/ .
Again, (iii) is equivalent to:
(iii)
for each x  X , supp ( x, G)  v V ; x  (v) is an independent set of vertices of
G.
4
Some might prefer to replace “  (v)   (u )  0/ ” in (iii) by “   (v)   (u )   0 ”,
but, as long as we are dealing with graphs of finite, or even countable, order, this annoying
relaxation makes no difference.
Given ( X , p ,  ), x  X , L,  and a subgraph H of G, let  ( x, L , H ) be defined
as it was in the preceding section; that is,  ( x, L, H ) is the vertex independence number of
 supp L ( x, H )  H . Note that  (  , L, H )  max S  vS  L(v ) , in which the maximum is
taken over independent subsets S of V(H), and  L ( v ) stands for the characteristic function
of L(v)  p ; thus  (  , L, H ) is a measurable function on X. (Indeed,  (  , L, H ) is a
non-negative measurable simple function, since it can take only the values 0, 1, …,  (H) =
the vertex independence number of H .)
We will say that (G, L, ) satisfies Hall’s condition if and only if, for each subgraph
H of G,

 (v ) 
vV ( H )
X  ( x , L, H ) d  ( x ) .
* (L, , H, )
Since if H1 is a subgraph of H obtained by deleting edges then  ( x, L, H )   ( x, L, H1 )
for all x  X , it is clear that (G, L,  ) satisfies Hall’s condition if and only if * (L, , H,
) holds for all induced subgraphs H of G.
Proposition 1. Hall’s condition is necessary for the existence of a proper (L, )-colouring
of G.
Proof: Suppose that  is a proper (L, )-colouring of G. Suppose that H is a subgraph of
G. Because  satisfies (iii), above,  (  ,  , H )   vV ( H )  ( v ) ; because  (v)  L(v) for
each v V ,  (  ,  , H )   (  , L, H ) . Therefore,

vV ( H )
 (v ) 
   (v) 

X  ( x,  , H )d  ( x)

X  ( x, L, H )d  ( x) .
vV ( H )

5
Following the terminology of Fremlin [4], we will say that ( X , p ,  ) is atomless
and semifinite if and only if for each Ap
with  ( A)  0 , there exists B  A
satisfying 0   ( B)   ( A) .
All such spaces in this paper are non-trivial, meaning  ( X )  0 . It is an ancient
result of Carathéodory (see [4]) that an atomless, semifinite positive measure space
( X , p ,  ) satisfies the following apparently more stringent requirement, which we shall
occasionally refer to as arbitrary divisibility: if A  p ,  ( A)  0 , and 0  r   ( A) , then
there exists B  A satisfying  ( B)  r . It follows that for such measure spaces, it would
not matter if the requirement of (ii) in the definition of proper (L, )-colouring were changed
from   (v)    (v) to   (v)    (v) .
Now we give two definitions and a result that will prove useful in stating and proving
our main result, in the next section, which is an analogue of Theorem GCL for atomless,
semifinite measure spaces.
Definition of CGL( ) . For an arbitrary positive measure space (X, p , ), let CGL( )
denote the collection of graphs G such that whenever L : V  p
and  : V  [0, )
satisfy Hall’s condition, with G, then there is a proper ( L,  )-colouring of G. (Note that in
this definition we are not allowing  as a value of . We leave to others, for now, the
pleasure of investigating what happens if  (v)   is allowed.)
Next we retrieve from [2] the definition of k
Definition of k
0
. k
0
0
.
denotes the collection of graphs G in which every block of G is
a clique, and every cut vertex of G is in exactly two blocks.
Thus k
0
is the collection of graphs characterized in Theorem GCL.
Proposition 2. Suppose that G  CGL(  ) and G1 is an induced subgraph of G. Then
G1  CGL( ) .
6
Proof: Suppose that G1  CGL(  ) . Then there exist L : V (G1 )  p
and
 : V (G1 )  [0, ) such that G1 , L, and  satisfy Hall’s condition, yet there is no proper
(L, )-colouring of G1 . Extend L and  to all of V  V (G ) by setting L(v)  0/ and
 (v)  0 for v V \ V (G1 ) . Since G1 has no proper (L, )-colouring, G cannot have one
either. We show that G, L and  satisfy Hall’s condition, which will prove that
G  CGL(  ) .
Suppose that H is an induced subgraph of G. Let H1 be the subgraph of G
induced by V ( H )  V (G1 ) . Because G1 is an induced subgraph of G, H1 is a subgraph of
G1 . Thus, because G1 , L, and  satisfy Hall’s condition, we have

vV ( H )
 (v ) 

 (v )

X  ( x, L, H1 )d  ( x)

X  ( x, L, H )d  ( x) .
vV ( H1 )

In other versions of Proposition 2, where assignments such as those giving the
extensions of L and  in the proof are not allowed, define L(v) to be a set of relatively
huge -measure, and  (v) to be relatively small, for each v  V \ V (G ) . A trick of this
sort is used in the proof of Proposition 3, below.
Finally we remark that in the measurable spaces that are of interest in this paper
(atomless, semifinite and positive), it makes no difference whether the “=” in (ii) is replaced
by “”. However, for other measure spaces it does make a difference. For example, consider
the case where  is the counting measure on an infinite set X. The usual list multicolouring
described in the first section can be regarded as a special case of (X, , ) list
multicolouring, in which X = C,  is the collection of all subsets of C (even though only
finite subsets are assigned), and  is the counting measure. In the first section,  is
required to be integer valued, but here and in the rest of the paper  is only required to be
real and non-negative. If X = C,  is the counting measure, and “=” in (ii) is replaced by
“”, then there is a proper (L, )-colouring of G if and only if there is a proper
 L,    -
colouring of G. But with  taking non-integral values, Theorem CGL is not true even with
the less severe definition () of a proper (L, k)-colouring. To see this, take G  K 2 ,   12
7
and let L assign the same singleton to both vertices of G. Then Hall’s condition is satisfied,
yet no proper (L, )-colouring is possible. The foundation of this irritating example becomes
clear when one observes that
 G, L,   
does not satisfy Hall’s condition.
We also notice that if  is the counting measure, then k
0
is not CGL() [cf. the
main result, below]. In fact, by ploughing through the definitions carefully, we can see that
CGL( )   if we have (ii) with “=”, whereas CGL( )  K n : n  1, 2, 3, ... if we have
(ii) with “” (where K n denotes the graph with n vertices and no edges).
3.
The Main Result
Theorem. If ( X , p ,  ) is atomless and semifinite, then CGL( )  k
Proof: First we shall show that CGL(  )  k
0
0.
by showing that none of Ck , k  4 , nor
K 4 -minus-an-edge, nor K1, 3 is in CGL(  ) . The inclusion will then follow from
characterization (d) in Theorem CGL of k
0,
and from Proposition 2.
G  K 4 -minus-an-edge: By arbitrary divisibility, there exist disjoint sets
A, B, C  X with equal positive finite measures – say  ( A)   ( B)   (C )  z  (0, ) .
Consider G with   z and with the list assignment L indicated in Figure 1.
AC
A B
C
B C
Figure 1
It is clear that no proper (L, )-colouring of G is possible. To see that (G, L,  )
satisfies Hall’s condition, first verify that G  v is properly (L, )-colourable for each
8
v V ; by Proposition 1 this verifies  ( L,  , H ,  ) for each proper induced subgraph H of
G. For the case H = G, observe that
X  ( x , L, G ) d  ( x )
  ( A)   ( B)  2  (C )
 4z 
  (v ) .
vV
G  Ck , k  4 : The L and  preceding work for C4 , and for k > 4 the idea is
similar. Let u, v be vertices on Ck a distance  2 from each other in both directions
around the cycle. Let the vertices from u to v along one arc of the cycle be w1, ..., ws , and
along the other arc x1 , ..., xt . Note that k  s  t  2 , and s, t  1 . Let A1, ..., As , B1 , ..., Bt
and C be disjoint measurable subsets of X with equal positive, finite measure z. Set   z
and define L(u )  A1  B1 , L( wi )  Ai  Ai 1 , 1  i  s 1 , L( s )  As  C ,
L( xi )  Bi  Bi  1 , 1  i  t 1, L( xt )  Bt  C , and L(v)  C . No proper (L, )-colouring
is possible, for in any such colouring v would have to be coloured C minus a set of
measure zero; this would then force colouring wi with Ai , i  1, ..., s and xi with
Bi , i  1, ..., t , except for sets of measure zero, leaving no possibility for colouring u with a
subset of A1 B1 of any positive measure. To verify that Hall’s condition is satisfied, as in
the previous case, observe that G  y is properly (L, )-colourable for each y  V , and
verify  ( L,  , G,  ) directly by noting that  ( x, L, G )  1 if x  A1  ...  As  B1  ...  Bt
and  ( x, L, G )  2 if x  C , so
X  ( x, L, G)d  ( x)  ( s  t  2) z  kz  yV  (v) .
G  K1,3 : Let A, B, C be disjoint measurable subsets of X of equal positive finite
measure z . With u, v, w and x as in Figure 2, set  (u )  2 z ,  (v)   ( w)   ( x)  z ,
L(u )  A  B  C , L(v)  A  B , L( w)  A  C , and L( x)  B  C .
v
u
w
x
Figure 2
9
This is a case in which it is quite easy to see that Hall’s condition is satisfied, but not
entirely trivial to see that no proper (L, )-colouring of G is possible. To see this, suppose
that  is a proper (L, )-colouring of G ; let A1   (u )  A , B1   (u)  B , and
C1   (u )  C . Then  ( A1 )   ( B1 )   (C1 )  2 z , so two of the three numbers  ( A1 ) ,
 ( B1 ) ,  (C1 ) sum to a number  4 z 3 . Without loss of generality, assume that
 ( A1)   ( B1 )  4 z 3 . But then  (v)  ( A  B) \ ( A1  B1 ) , so
  (v)   2 z  4 z 3  2 z 3  z   (v) . Thus  is not a proper (L, )-colouring of G, after
all.
Now we set about proving the hard part of the theorem, k
0
 CGL( ) . We shall
adapt the brilliant proof of Bollobás and Varopoulos [1] and use Theorem CGL as a sort of
foundation. (The result in [1] is just what we are trying to prove here, in the case where G is
a clique, possibly infinite.)
Suppose that G  k
0
and that G with L : V  p
and  : V  [0, ) satisfies
Hall’s condition. Colour any vertices at which   0 with 0/ and delete them from G ;
the resulting graph is still in k
0.
Thus we may suppose that all the values of  are
positive. We may also suppose that   L(v)    for each v V , for if   L(v)    , by
arbitrary divisibility we can find a measurable subset A of L(v) with
uV  (u)   ( A)  
and replacing L(v) with A gives a new situation in which Hall’s
condition is still satisfied; and a proper colouring with respect to the new list assignment will
serve as a proper colouring with respect to the old.
Let L ( X ,  ) denote the space of equivalence classes, under “almost everywhere
agreement”, of the essentially bounded -measurable real-valued functions on X. We will
follow the conventional practice of referring to these equivalence classes as functions.
L ( X ,  ) is a Banach space with the norm

= essential supremum of the absolute
value, and, as such, is isometric to the dual of the Banach space commonly denoted
L1 ( X ,  ) , via the pairing  ,     d  ,   L1 ( X ,  ) ,   L ( X ,  ) . Let B denote
X
the closed unit ball of L ( X ,  ) .
10
The Tychonoff-Alaoglu theorem implies that B is compact in the “weak-” topology
on L ( X ,  ) , thought of as the dual of L1 ( X ,  ) ; this topology is also known as the
topology of L1 ( X ,  ) pointwise convergence on L ( X ,  ) . It follows that BV is compact
in L ( X ,  )V , equipped with the product topology  arising from the weak- topology on
L ( X ,  ) . For   0 let F ( )  BV consist of the functions  : V  B satisfying:
(F1)
For each v V ,  (v ) is non-negative;
(F2)
for each v V , supp  (v)   x  X ;  (v)( x)  0  L(v) ;
(F3)
for each block D of G, 

vV ( D )
(F4)
for each v V ,
(v)   1 ; and
X (v)d    (v)  
.
It is an elementary exercise to see that F ( ) is a -closed subset of BV . (That
  L(v)    for each v V plays a role in this verification.) Also, it is clearly convex,
and 0  1   2 implies F (1 )  F ( 2 ) . These facts will play a role later.
Next we shall use Theorem CGL to show that for each   0 there is some
  F ( ) such that
(1)
for each v V ,  (v ) is the characteristic function of some measurable set
 (v )  L (v ) ;
(2)
  (v)    (v)   for each v V ; and
(3)
if u, v V are adjacent in G, then  (u )   (v)  0/ .
Note that (1) implies that   BV and satisfies (F1) and (F2); (2) and (1)
imply (F4); finally, (1) and (3) ensure that for any block D of G, a clique because
Gk
0,
and any x  X ,
 vV ( D) (v)( x)  1 ,
which establishes (F3). Thus (i ) ,
i  1, 2, 3 imply   F ( ) .
Suppose 0    min vV  (v) . Considering the Venn diagram of the sets L(v) ,
v V , meaning the sets of the form
L(v) ,   U  V , we can find pairwise
L(u ) \
uU
vV \U
11
disjoint sets of positive measure, A1, ..., A t , 1  t  2|V | 1 , such that each L(v) is the
disjoint union of some of the Ai , except possibly for a set of measure zero.
Let r (v)  s(v) d be rational numbers with common denominator d  2t  such
that  (v)  2  r (v)   (v) for v V . By arbitrary divisibility, each Ai is the disjoint
union Bi1  ...  Bipi  Ci of measurable sets such that  ( Bi j )  1 d , 0   (Ci )  1 d and
0   (Ci )  1 d , i  1, ..., t , 1  j  pi . We introduce symbols bi j in correspondence to the
Bi j and set L(v)  bi j ; Ai  L(v), 1  j  pi  and  (v)  s(v)  t . [Note that
t d   2   (v)    2   r (v)  s (v) d , so s (v)  t is a positive integer.]
We claim that G, L , and  satisfy Hall’s condition, in the discrete case, with
reference to  ( L,  , H ) . To see this, suppose that H is a subgraph of G. For v V , let
I (v)  i ; Ai  L(v) and, possibly modifying the sets L(v) by removing subsets of
measure zero, assume that L(v) 
iI ( v )
A i for each v V . Because A i is either
contained in L(v) or disjoint from it, for each v V , it follows that  ( x, L, H ) is constant
on A i ; say  ( x, L, H )  ai , x  Ai , i  1, ..., t . Since bi j  L(v) if and only if Ai  L(v)
(i.e. i  I (v ) ), we also have that
 (bi j , L, H )  ai , 1  j  pi , 1  1, ..., t .
We have
1
 s (v )
d vV ( H )


r (v ) 
vV ( H )


 (v )
vV ( H )
X  ( x, L, H )d 
[because (G, L, ) satisfies Hall’s
condition, by assumption]


i Ai
 ( x , L, H ) d 
[since  ( x, L, H )  0 for
x

vV ( H ) L(v )

i
t
 A  ( x, L, H )d 
i 1
t
1
  ai  ( Ai ) 
i 1
[the Ai are disjoint]
t
 ai ( pi  1)
i 1
d
Ai ]
12
1 t
 ai pi  t | V ( H ) | d ,
d i 1

whence

vV ( H )
Since G  k
Define  : V  p

 s (v )  t  
 (v ) 
vV ( H )
0,
t
 ai pi 
i 1
t
pi
 (bi j , L, H ) .
i 1 j 1
by Theorem CGL there is a proper ( L,  ) -colouring  of G.
by  (v)  (i , j ); bi j  ( v ) Bi j and  by  (v)   ( v ) .
If bi j  (v)  L(v) then i  I (v ) , i.e. Bi j  Ai  L(v) ; thus  (v)  L(v) . If u, v
are adjacent in G then  (u )   (v)  0/ , so  (u )   (v)  0/ , since the Bi j are pairwise
disjoint. Thus (1) and (3) hold. As for (2),
  ( v )  

(i , j ); bi j  ( v )
 ( Bi j )   s(v)  t  d
 r (v )  t d
  (v )   2   2   ( v )   .
Before moving on to the last stage of the proof, it is worth noting that the preceding
gives a proof of this part of the Theorem, that G  k
0
and G, L,  satisfying Hall’s
condition imply the existence of a proper (L, )-colouring of G, under the stronger
hypothesis that each of the inequalities  ( L,  , H ,  ) is strict. Indeed, if each of those
inequalities is strict then, because there are only finitely many of them, and | V | is finite,
then  ( L,    , H ,  ) holds for all H  G , for some small   0 . It then follows that
there is a proper colouring  : V  p
with  (v)  L(v) and  (v)   (v) , v V . We
can then arrange that  (v)   (v) , v V , by arbitrary divisibility.
The interest in this proof is the promise it holds of practical feasibility in this proof.
Following it, we see that, given G, L, and  with each inequality  ( L,  , H ,  ) satisfied
strictly, a proper (L, )-colouring  of G can actually be constructed, or found, if
(i)
arbitrary divisibility is feasible – in particular, given A i and d, we need to find sets
Bi j , 1  j  pi , and Ci , of which A i is the disjoint union, with  ( Bi j )  1 d and
 (Ci )  1 d ; and
13
(ii)
Theorem CGL can be implemented – in this case, to find the colouring  . (Of
course, Theorem CGL can be implemented, by brute force, since finding a colouring
in the discrete case involves checking “only” a finite number of possibilities, but the
need for practical feasibility includes a need for efficiency.)
We now have that the sets F ( ) are non-empty, -closed subsets of BV , -
compact, and are nested; therefore F 
 0 F ( )
is non-empty. It is also -closed and
convex, and thus -compact and convex. Since  is a locally convex, Hausdorff topology, it
follows from the Krein-Milman theorem that F is the -closure of the convex hull of its
extreme points. Since F is non-empty, it follows that F has an extreme point .
Every member of F satisfies (F1), (F2), and (F3), and (F4) with  (v)   replaced
by  (v) , and every member of BV satisfying these requirements is in F. Suppose that,
for each v V ,  (v) is the characteristic function of some  (v)  p . Then, for each
v V  (v)  L(v) ,   (v)    (v) , and if u is adjacent to v then u and v lie in a
common block of G, so by (F3)
 (u )   (v)

 1 , whence   (u )  (v)   0 .
Therefore  can be trimmed down to a proper (L, )-colouring of G.
We will finish the proof by showing that if  (v) is not a characteristic function for
every v V , then  is not an extreme point of F after all. We will do this by finding
(assuming not every  (v) is a characteristic function) a set Y  X of positive measure, a
positive number  , and a path u1 , ..., um in G satisfying
(i)
  (ui )  1   on Y, i  1, ..., m ;
(ii)
in case m > 1, for i  1, ..., m  1 , ui and ui  1 are in the same block Di of G,
and the blocks D1, ..., Dm1 are distinct;
letting
(iii)
Y
f
be defined by
Y
 f  Y

,
if u1 is in a block D0 different from D1 (which can only happen if u1 is a cutvertex of G), then 

vV ( D0 )
 (v) Y  1   ,
and if um is in a block Dm different from Dm 1 , then 

vV ( Dm )
 (v) Y  1   ;
14
(iv)
if m = 1 and D is a block of G containing u1 (there are at most two such blocks),
then


vV ( D )
The assumption that G  k
0
 (v) Y  1   .
and (ii) imply that if m  3 , then u2 , ..., um1 are
cut-vertices, and each block Di , 1  i  m 1 , contains only two points of the path, ui and
ui  1 . It is important to observe that if any of the inequalities in (i)-(iv) hold, they will
continue to hold if  is replaced by a smaller positive number, and if Y is replaced by a
subset of Y with positive measure. In what follows, both  and Y will be steadily
shrinking, without name changes.
Before going on to produce Y,  , and u1 , ..., um , let us suppose that we have them,
and that (i)-(iv) hold. By arbitrary divisibility Y can be partitioned into two sets, Y1 and
Y2 , of equal measure  (Y ) 2  0 . We define 1 and  2 by
 i   on V \{u1, ..., um} , i  1, 2 ,
  (ui )   ( Y1  Y2 ) ,

1 (ui )  
  (ui )   ( Y2  Y1 ) ,
i odd
  (ui )   ( Y2  Y1 ) ,

 2 (ui )  
  (ui )   ( Y1  Y2 ) ,
i odd
i even ,
i even .
Because  > 0 and Y1 , Y2 have positive measure, 1   2 , and clearly   12  1   2  .
So, if 1 ,  2  F , we are done.
It is clear that 1, 2  BV by the way they are defined, and (i), and (F1) holds, for
the same reason. Again invoking (i), supp i (v)  supp (v) , i  1, 2, v V , so (F2)
holds. That Y1 , Y2 have equal measure implies that, for each v V ,
X i (v)d 

and thus the  i satisfy (F4), with   0 .
X (v)d  ,
i  1, 2,
15
Finally, we will see that 1 and  2 satisfy (F3). 1 and  2 are obtained by
modifying the values of  (v) only by small amounts, only on Y, and only for
v {u1 , ..., um } . If D is a block of G, then either D contains none of the ui , or only one
of the ui , or two of the ui . If none, there is no difficulty; if only one, then by (iii) and (iv)
(if m = 1),


vV ( D )
 (v) Y  1   ,
which implies, for i = 1, 2,


vV ( D )
 i (v)   max 

 i (v) Y , 
vV ( D )
 max 


vV ( D )
 (v) Y   , 
v V ( D )
 i (v) X \Y 

 (v)  
v V ( D )
 max 1     , 1  1.
If D contains two of the ui , then for some 1  i  m  1 , D  Di and D contains
ui and ui  1 , and no other u j . But because ui and ui  1 are adjacent members of the
path, by the way the  i are defined, we then have

vV ( D )
 i (v ) 

 (v) , i  1, 2,
vV ( D )
which again implies that 1 ,  2 satisfy (F3).
So, it remains only to find Y,  , and u1 , ..., um , given that  (v) is not a
characteristic function, for some v  V . We can hunt among the vertices v such that  (v)
is not a characteristic function (a.e.) and find one such, u1 , such that either u1 is not a
cut-vertex, or, if u1 is a cut-vertex and thus in exactly two blocks of G, for all vertices v
on at least one “side” of u1 , i.e. in D  u , for one of the two blocks D in which u1
resides,  (v) is a characteristic function (a.e.).
That (u1 )  B , non-negative, is not a characteristic function (a.e.) means that
0  (u1 )  1 on a subset of L(u1 ) of positive measure. By standard arguments there exists
a set Y  L(u1 ) of positive measure and a number   0 such that
  (u1 )  1   on Y .
16
Because  satisfies (F3), and (u1 )   on Y, for any block D of G containing
u1 ,

 (v)  1   on Y. Therefore, if v  D and  (v) is a characteristic function,
vV ( D )
v u1
then  (v)  0 on Y (a.e.). Therefore, if u1 is a cut-vertex of G and D is a block
containing u1 with  (v) a characteristic function for all v V ( D) \{u1} , then  (v)  0
on Y, a.e., for all v V ( D) \{u1} , so 

(v) Y  (u1 ) Y  1   .
vV ( D )
Whether or not u1 is a cut-vertex of G, let D1 be the block of G containing u1
which might possibly contain other vertices v such that  (v) is not a characteristic
function. If 

(v) Y  1 , set m = 1 and replace  by min  , 1  

vV ( D1)

 (v) Y  ;

v V ( D1)
we are done.
So suppose that 

 (v) Y  1 . (Because  satisfies (F3),
vV ( D1 )


 (v) Y  1 .) Since (u1 )  1   on Y, there must be some u2 V ( D1 ) such that
vV ( D1 )
(u2 )  0 on a subset of Y of positive measure. We have already noted that by (F3), and
because  (u1 )   on Y, and because u1 and u 2 are in the same block, (u2 )  1  
on Y. By standard arguments we can achieve   (u2 )  1   on Y, with old 
possibly replaced by a new smaller   0 , and the old Y possibly replaced by a subset of it
with positive measure.
If u 2 is not a cut-vertex of G, take m = 2 and we are done. Otherwise, let D2 be
the block of G other than D1 containing u 2 . If 

 (v) Y  1 replace  by
vV ( D 2 )
min  , 1  


 (v) Y  , take m = 2, and we are done. Suppose that

v V ( D2 )


 (v) Y  1. By arguments advanced when we were in this case with u1 and D1 ,
vV ( D 2 )
there must be some u3 V ( D2 ) \{u2 } such that 0  (u3 )  1   on a subset of Y of
positive measure. Again, by standard arguments, we can assume that   (u3 )  1  
17
on Y, for a possibly smaller   0 and a new Y, a subset of the old Y, still of positive
measure. If u3 is not a cut-vertex of G, take m = 3 and we are done. Otherwise, on to the
next block, D3 , the block on the other side of u3 from D2 .
Since G is finite, this process, if we have got this far, will end at some um , m  3 ,
which is in the same block Dm 1 of G as um 1 , and which is either not a cut-vertex of G
(in which case we are done), or, if it is a cut-vertex, then the block Dm containing um
other than Dm 1 satisfies 

 (v) Y  1 . In this case replace the current value of 
vV ( D m)
by min  , 1  


 (v) Y  , and we are done.


v V ( Dm )
There is one other fundamental piece of unfinished business concerning list
multicolourings with measurable sets from atomless, semifinite measure spaces that we are in
a position to settle now. Cropper, Gyárfás, and Lehel, in [3], actually prove a stronger
theorem than Theorem CGL, a result which , when combined with a result in [8], enables us
to consider graphs with cut-vertices which are in more than two blocks.
Theorem CGL. Suppose that G is a simple graph and  : V (G )  {1, 2, ...} . Then the
pair (G, ) has the property that there is a proper (L, )-colouring of G for every
assignment L of finite sets to V (G ) which satisfies Hall’s condition with G and  if and
only if every block of G is a clique and for every cut-vertex v of G in more than two
blocks,  (v )  1 .
Evidently because there is greater freedom in devising list assignments when one is
assigning measurable sets from atomless measure spaces, the analogue of Theorem CGL in
the world of such assignments is much less interesting than Theorem CGL.
Proposition 3. Suppose that G is a simple graph,  : V (G )  (0, ) , and ( X , p ,  ) is
a positive, atomless, semifinite measure space with  ( X ) 

vV ( G )
 (v) . If G  k
0
then
18
there is a list assignment L : V (G )  p
which satisfies Hall’s condition with G and ,
but such that there is no proper (L, )-colouring of G.
From the theorem of this section it then follows that the pairs (G, ), as above, for
which the satisfaction of Hall’s condition by a list assignment L, as above, is sufficient for
the existence of a proper (L, )-colouring, are just the pairs for which G  k
0
. That is, 
has nothing to do with it. Thus the “atomless” case differs from the discrete case in this
respect.
Proof of Proposition 3. Suppose that ( X , p ,  ) , G, and  are as in the proposition
statement. Since G  k
0
then G contains at least one of Ck , for some k  4 , K 4 -
minus-an-edge, or K1,3 as an induced subgraph, which we will call H. If we can find a list
assignment L : V ( H )  p
which satisfies Hall’s condition with H and , but such that
there is no proper (L, )-colouring of H, then we can extend L to V(G) by assigning a set
of measure

 (v) to each vertex in V (G ) \ V ( H ) to obtain a list assignment which
vV ( G )
satisfies Hall’s condition with G and , but such that there is no proper (L, )-colouring of
G.
The awkward, uncolourable-from list assignments to Ck , k  4 , K 4 -minus-anedge, and K1,3 , are inspired by those given in the early, “easy” part of the proof of the
theorem; but in that proof we had the luxury of specifying  as well as L, whereas here 
is given, and we have no control over it. When the reader sees how we overcome this
difficulty in one case, it will be clear how to proceed in the other cases, in light of the earlier
proof. So we will end the proof by giving the argument in only one case, that of K1,3 .
Let the vertices of K1,3 be labelled as in Figure 2. Let r > 0 be small enough so
that 2r   (u ) , and r  min  (v),  ( w),  ( x)  . By arbitrary divisibility, there exist
pairwise disjoint subsets of X ,
A , B , C , D(u ), D(v), D( w), D( x), satisfying
 ( A)   ( B)   (C )  r ,
19
  D(u )    (u )  2r , and
  D( y )    ( y )  r , y  v, w, x .
Then L is defined by
L(u )  A  B  C  D(u ) ,
L ( v )  A  B  D (v ) ,
L( w)  A  C  D( w) , and
L( x)  B  C  D ( x) .
We leave it to the reader to verify that K1,3 , L, and  satisfy Hall’s condition, and that
there is no proper (L, )-colouring of K1,3 .

References
1.
B. Bollobás and N.Th. Varopoulos, Representation of systems of measurable sets,
Math. Proc. Camb. Phil. Soc. 78 (1974), 323-325.
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M.M. Cropper, J.L. Goldwasser, A.J.W. Hilton, D.G. Hoffman, and P.D. Johnson Jr.,
Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to listmulticolorings of graphs, J. Graph Theory 33 (2000), 199-219.
3.
M. Cropper, A. Gyárfás, and J. Lehel, Edge list multicoloring trees: an extension of
Hall’s theorem, J. Graph Theory 42 (2003), 246-255.
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D.H. Fremlin, Measure Theory, Vol. 2: Broad Foundations, Torres Fremlin, 2002.
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P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26-30.
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P.R. Halmos and H.E. Vaughan, The marriage problem, Amer. J. Math. 72 (1950),
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A.J.W. Hilton and P.D. Johnson Jr., An elementary proof of Hall’s selection principle
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P.D. Johnson Jr. and E.B. Wantland, Two problems concerning Hall’s condition,
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R. Rado, A theorem on general measure functions, Proc. London Math. Soc. 44
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