Circular Chromatic Index of Regular Graphs with
Maximum Degree Greater than Three
Barbora Candráková
Comenius University in Bratislava
Faculty Of Mathematics, Physics And Informatics
Advisor: RNDr. Ján Mazák
23.6.2011
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Edge Coloring
An r -edge-coloring of a graph G
It is a mapping c : E (G ) → {0, 1, 2, . . . , r − 1} satisfying
1 ≤ |c(e) − c(f )| ≤ r − 1
for any adjacent edges e and f of a graph G .
χ0 (G ) = min{r ; ∃ r -edge coloring of G}.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Edge Coloring
An r -edge-coloring of a graph G
It is a mapping c : E (G ) → {0, 1, 2, . . . , r − 1} satisfying
1 ≤ |c(e) − c(f )| ≤ r − 1
for any adjacent edges e and f of a graph G .
χ0 (G ) = min{r ; ∃ r -edge coloring of G}.
A 4–regular snark
A 4–regular snark is a 4–regular graph with chromatic index five.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Circular Edge Coloring
A circular r -edge-coloring of a graph G
It is a mapping c : E (G ) → [0, r ) satisfying
1 ≤ |c(e) − c(f )| ≤ r − 1
for any adjacent edges e and f of a graph G .
χ0c (G ) = inf {r ; ∃ circular r -edge coloring of G}.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Circular Edge Coloring
A circular r -edge-coloring of a graph G
It is a mapping c : E (G ) → [0, r ) satisfying
1 ≤ |c(e) − c(f )| ≤ r − 1
for any adjacent edges e and f of a graph G .
χ0c (G ) = inf {r ; ∃ circular r -edge coloring of G}.
A (p, q)-coloring
A (p, q)-coloring of a graph G is a mapping
c : E (G ) → {0, 1, 2, ..., p − 1} satisfying
q ≤ |c(e) − c(f )| ≤ p − q
for any adjacent edges e and f , p and q are positive integers.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Properties of the Circular Coloring
Theorem
For every graph G ,
χ0 (G ) − 1 < χ0c (G ) ≤ χ0 (G ).
0
4
2
1
0
1
2
3
4
3
4
2
0 5/2
3
2
1
0
1
7/2
0
1
0
1
7/2
5/2
2
3/2
3
1/2 4
3/2
1/2
2
3
4
Figure: A 4–regular snark. χ0 (G ) = 5, χ0c (G ) = 9/2.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Properties of the Circular Coloring
p
q
a (p, q)-coloring corresponds to a (p/q)-coloring
χ0c (G ) =
the circular chromatic index is rational
the circular chromatic index is always attained
n
o
χ0c (G ) = min pq : there exists a (p, q)-coloring of G .
p ≤ |E (G )|
q ≤ |V (G )|/2
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Aims of the Thesis
to study circular colorings of d-regular snarks – d-snarks
to establish lower bounds on the circular chromatic index of
d-snarks
to determine values of circular chromatic index for some
classes of d-snarks, especially 4-snarks
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Related Work and Motivation
circular colorings of cubic graphs
some results for special classes of d-regular graphs, but not
for general ones (West, Zhu, 2008)
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Related Work and Motivation
circular colorings of cubic graphs
some results for special classes of d-regular graphs, but not
for general ones (West, Zhu, 2008)
cubic graphs - cycle double cover conjecture
line graph of a cubic graph G is 4-regular, if G is snark, then
L(G ) is snark
circular colorings can be used in scheduling problems
(Modarres, Ghandehari 2008), (Yeh, Zhu 2005)
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
A Lower Bound on the Circular Chromatic Index
NP-hard to determine the value of the circular chromatic index
a lower bound can help to exclude some possible index values
depending on the order of the graph (no. of vertices)
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
A Lower Bound on the Circular Chromatic Index
Theorem
Let G be a d-regular snark on 2k vertices, with k > 2. Then
χ0c (G ) ≥ d +
2
.
k −1
trivial bound for p, q
d-regular graph is a snark if and only if it doesn’t have d − 1
disjoint perfect matchings (1-factors)
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
A Lower Bound on the Circular Chromatic Index
Theorem
Let G be a d-regular snark on 2k vertices, with k > 2. Then
χ0c (G ) ≥ d +
2
.
k −1
trivial bound for p, q
d-regular graph is a snark if and only if it doesn’t have d − 1
disjoint perfect matchings (1-factors)
this bound is tight for d-regular snarks – we can construct a
2
class of snarks with χ0c (G ) = d +
k −1
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
4-snarks with χ0c = 4 +
2
k−1
the class {Gm }∞
m=1
Gm : we take 2 cycles of length 2m + 1, 2m parallel edges, 1
simple edge, join together by 2 simple edges
Figure: The graph G2 .
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Lower Bound for 4-snarks with Additional properties
Lower bound can be improved if we put some constraints on
certain properties of graph.
Theorem
Let G be a 4-regular snark on 2k vertices with a girth greater than
4. Then
5
2.5
χ0c (G ) ≥ 4 +
=4+
.
2k
k
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Lower Bound for 4-snarks with Additional properties
Lower bound can be improved if we put some constraints on
certain properties of graph.
Theorem
Let G be a 4-regular snark on 2k vertices with a girth greater than
4. Then
5
2.5
χ0c (G ) ≥ 4 +
=4+
.
2k
k
Theorem
Let G be a d-regular snark on 2k vertices, with k > 2. Let G have
no 1-factors. Then
χ0c (G ) ≥ d +
Barbora Candráková
d
.
k −1
Circular Chromatic Index of Regular Graphs with Maximum De
Construction of 4-snarks with χ0c (G ) = 4 + 2/3m, m ≥ 2
a
b
d
c
Figure: A building block.
u
...
B1
Bm
B2
...
Figure: A class of 4-snarks.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Construction of 4-snarks with χ0c (G ) = 4 + 2/3m, m ≥ 2
,2
+
,1
+
ε]
]
2ε
[3,3+ε]
[3,3+2ε]
2ε
]
[2
β ∈ [-2ε,2ε]
[1
+
,1
-ε
[1
ε]
α=0
[2,2+2ε]
[2-ε,2+2ε]
[3
-ε
,
3+
[1-ε,1+ε]
[-ε,ε]
[2
-ε
,2
+
ε]
ε]
ε,
[-
δ ∈ [1-ε,1+ε]
[-
ε,
ε]
]
2ε
+
,3
[3
γ ∈ [1-ε,1+2ε]
Figure: A coloring by circular intervals of the building block.
small change of colors on dangling edges
for this building block the maximal change of colors on
dangling edges is 3ε
m blocks, 2 ≤ m · 3ε, ε ≥ 2/3m
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Construction of 4-snarks with χ0c (G ) = 4 + 2/3m, m ≥ 2
3
9m+2
6m
+
2
3
+
3m
3m
+
2
0
6m+3
1
9m+3
9m+4
1
+
3m
6m+1
2
+
2
3
+
6m
1
3m
9m+3
6m+4
Figure: A (12m + 2, 3m)-coloring of the block.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Construction of 4-snarks with χ0c (G ) = 4 + 2/r , r ≥ 3
similar to the previous construction
using two types of building blocks, change of color 3ε and ε
a
b
d
c
join certain number of eight-blocks in a row, add a few
six-blocks, depending on the value of r
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Conjecture for d-regular Graphs
Conjecture [Afshani et. al., 2005]
For any integer k > 1 there is an ε > 0 such that no graph G has
k − ε < χ0c (G ) < k.
if the conjecture is true, what is the value of εk ?
is it k − 1/(k − 1)? (holds for k = 2, 3, 4)
k = 5 – find 4-regular snark with circular chromatic index
5 − 1/4 = 4 + 3/4 = 19/4.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
The Circular Chromatic Index of 4-snarks with Girth ≥ 3
order
5
6
7
8
9
10
11
12
13
no. of graphs
1
1
2
3
16
59
265
1544
10778
no. of 4-snarks
1
0
2
0
16
1
265
9
10778
Barbora Candráková
circ. chrom. index
≤ 9/2 14/3
5
1
2
12
2
2
1
261
1
3
9
10736
1 41
Circular Chromatic Index of Regular Graphs with Maximum De
The Circular Chromatic Index of 4-snarks with Girth ≥ 3
order
14
15
16
no. of 4-snarks
197
805491
11113
Barbora Candráková
≤ 9/2
14
804539
5289
circ. chrom. index
(9/2, 14/3]
5
3
180
3
949
15 5809
Circular Chromatic Index of Regular Graphs with Maximum De
The Circular Chromatic Index of 4-snarks with Girth ≥ 3
order
14
15
16
no. of 4-snarks
197
805491
11113
≤ 9/2
14
804539
5289
circ. chrom. index
(9/2, 14/3]
5
3
180
3
949
15 5809
Conjecture
There exist no 4-regular graph G with 14/3 < χ0c (G ) < 5.
Barbora Candráková
Circular Chromatic Index of Regular Graphs with Maximum De
Thank you.