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• A proper k-coloring of a graph G is an labeling
f: V(G) {1,2,...,k} such that adjacent verti
ces have different labels. The labels are colors;
The vertices of one color form a color class.
• A graph G is k-colorable if G has a proper k-co
loring.
• The chromatic number of a graph G, denoted b
y , is the least k such that G is k-colorab
le.
)(G
• A equitable k-coloring of a graph G is an prop
er k-coloring f: V(G) {1,2,...,k} such that
||f -1(i)|-|f -1(j)|| 1 for all 1 i j k.
• A graph G is equitably k-colorable if G has a e
quitable k-coloring.
• The equitable chromatic number of a graph G,
denoted by , is the least k such that G is
equitably k-colorable.
• The equitable chromatic threshold of a graph
G, denoted by , is the least k such tha
t G is equitably n-colorable for all n k.
)(G
)(* G
• If graph G is equitably k-colorable, then the size of al
l color classes in a nonincreasing sort will be
• or the sizes of all color classes in a nondecreasing sor
t will be
k
kGV
k
GV
k
GV 1)(...,,
1)(,
)(
,1)(
...,,1)(
,)(
k
kGV
k
GV
k
GV
• Conjecture. (Chen, Lih and Wu; 1994)
A connected graph G is equitable (G)-c
olorable if and only if
12,1212, nnnn KGandCGKG
• Theorem. (Guy; 1975)
A tree T is equitably k-colorable if k
• Theorem. (Bollobas and Guy ; 1983)
A tree T is equitably 3-colorable if
12
)(
T
10)(3)(8)(3)( TTVorTTV
• Theorem. (Chen and Lih ; 1994)
A tree T = T(X,Y), if and only
if
If , then
2)( T
.1|||| YX
2)( T .2)(* T
• Theorem. (Chen and Lih ; 1994)
Let T be a tree such that , then
, where v is an arbitrary major vertex.
2)( T
2])[\(
1||,3max)()( *
vNT
TTT
• Theorem. (Wu ; 1994)
is equitably k-colorable if and on
ly if
and for all i, where
tnnnK ,...,, 21
t
i
it
i
i
kn
nk
kn
n
11 //
k
n
kn
nni /
.1
t
iinn
• For n 2k+1, the Kneser graph KG(n,k) has
the vertex set consisting of all k-subsets of an
n-set. Two distinct vertices are adjacent in K
G(n,k) if they have empty intersection as subs
ets.
• Since KG(n,1) = Kn , we assume k 2.
1.
2.
3.
4.
1),(* knknKG
3),12(),12( * kkKGkkKG
65
7
2
1)2,()2,( *
n
n
if
if
n
nnKGnKG
137
1514
16
4
3
2
)3,()3,( *
n
n
n
if
if
if
n
n
n
nKGnKG
• Sketch proof of
• S is an i-flower of KG(n,k) if any k-subset in S contains
the integer i. An i-flower is an independent set of KG(n,
k).
• It is natural to partition the flowers to form an equitable
coloring of KG(n,k). Hence, if f is an equitable m-colori
ng of KG(n,k) such that every color class under f is con
tained in some flower, then m n-k+1.
1),(* knknKG
• KG(7,2) is equitable 6-colorable.
12
13
14
15
16
17
23
24
25
26
27
34
35
36
37
45
46
47
56
57
67Y:
C(7,2)=21=4+4+4+3+3+3
• KG(7,2) is equitable 6-colorable.
12
13
14
15
16
17
23
24
25
26
27
34
35
36
37
45
46
47
56
57
67Y:
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:
• KG(7,2) is equitable 6-colorable.
12
13
14
15
16
17
23
24
25
26
27
34
35
36
37
45
46
47
56
57
67Y:
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:
• KG(7,2) is equitable 6-colorable.
12
13
14
15
16
17
23
24
25
26
27
34
35
36
37
45
46
47
56
57
67Y:
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:
• KG(7,2) is equitable 6-colorable.
12
13
14
15
16
17
23
24
25
26
27
34
35
36
37
45
46
47
56
57
67Y:
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:
…
• Theorem. (P. Hall; 1935)
A bipartite graph G = G(X,Y) with bipart
ition (X,Y) has a matching that saturates every
vertex in X if and only if |N(S)| |S| for all S
X, where N(S) denotes the set of neighbors o
f vertices in S.
• KG(7,2) is equitable 6-colorable.
12
13
14
15
16
17
23
24
25
26
27
34
35
36
37
45
46
47
56
57
67Y:
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:
V1={12,15,16,17}, V2={24,25,26,27},V3={13,23,36,37},
V4={14,34,47}, V5={35,45,57},V6={46,56,67}