1Debajyoti
2Sue
Mondal
Whitesides
1University
2Rahnuma
3Md.
Islam Nishat
Saidur Rahman
of Manitoba, Canada
2University of Victoria, Canada
3Bangladesh University of Engineering and Technology (BUET), Bangladesh
Acyclic Coloring
6
6
1
11
5
2
2
4
1
4
3
1
6/21/2011
Acyclic Coloring of G
4
3
4
3
3 1
3
Input Graph G
1
45
1
4
3
1
IWOCA 2011, Victoria
2
Why subdivision?
6
6
1
11
5
2
2
4
Input Graph G
4
3
6/21/2011
1
4
3
1
4
3
3 1
Acyclic Coloring of
a subdivision of G
3
1
45
1
4
3
1
IWOCA 2011, Victoria
3
Why subdivision?
6
6
1
11
5
45
4 7
Division vertex
2
2 3
4
3 1
Acyclic Coloring of
a subdivision of G
3
Input Graph G
1
4
4
1
4
1
3
3
3
1
6/21/2011
4
3
3
1
IWOCA 2011, Victoria
4
Why subdivision?
Acyclic coloring of planar graphs
Upper bounds on the volume of 3-dimensional
straight-line grid drawings of planar graphs
Acyclic coloring of planar graph subdivisions
Upper bounds on the volume of 3-dimensional
polyline grid drawings of planar graphs
Division vertices correspond to the total number of
bends in the polyline drawing.
Input graph K5
A subdivision
G of K5
Straight-line
drawing of G in 3D
Poly-line drawing
of K5 in 3D
Previous Results
Grunbaum
1973
Lower bound on acyclic colorings of planar graphs is 5
Borodin
1979
Every planar graph is acyclically 5-colorable
Kostochka
1978
Deciding whether a graph admits an acyclic 3-coloring
is NP-hard
Ochem
2005
Testing acyclic 4-colorability is NP-complete for
planar bipartite graphs with maximum degree 8
Angelini &
Frati
2010
Every planar graph has a subdivision with one vertex
per edge which is acyclically 3-colorable
6/21/2011
IWOCA 2011, Victoria
6
Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
7
Some Observations
w3
3
w
v
3
w2 1
1 wn
w1 3
v
u 1
u 1
G
G
G/
G/
G/ admits an acyclic 3-coloring
6/21/2011
IWOCA 2011, Victoria
8
Some Observations
G is a biconnected graph that has a non-trivial ear decomposition.
2 m
Ear
1 n
3
g
1
a
2 l
f
2 b
1
Subdivision
2
3
h
k
c
j
2
1
e 1
d
2
i
x
3
2 l
G
G admits an acyclic 3-coloring with at most |E|-n subdivisions
6/21/2011
IWOCA 2011, Victoria
9
Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
10
Acyclic coloring of a 3-connected cubic graph
Subdivision
Subdivision
18
18
15
10
17
1
17
11
3
16
16
14
15 3
2
13
9
8
1
14
13
3
7
10 1 9
3
12
11
3
8
1 7
2 12
2
2 6
3
6
4
4
1
3
3
5
1
2
11
3
2
2
Every 3-connected cubic graph admits an acyclic 3-coloring
with at most |E| - n = 3n/2 – n = n/2 subdivisions
5
Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
12
Acyclic coloring of a partial k-tree, k ≤ 8
2 u
G
/
1
1
1
1
1
1
1
1
G
6/21/2011
IWOCA 2011, Victoria
13
Acyclic coloring of a partial k-tree, k ≤ 8
3 u
G
/
1
1
2
1
2
1
1
2
G
6/21/2011
IWOCA 2011, Victoria
14
Acyclic coloring of a partial k-tree, k ≤ 8
3 u
G
/
1
1
1
2
2
2
3
3
G
Every partial k-tree admits an acyclic 3-coloring for k ≤ 8
with at most |E| subdivisions
6/21/2011
IWOCA 2011, Victoria
15
Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
16
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
17
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
18
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
19
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
20
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
3
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
21
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
3
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
22
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
3
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
23
Acyclic 3-coloring of triangulated graphs
3
6
1
8
7
3
1
1
4
5
3
3
3
1
6/21/2011
2
1
IWOCA 2011, Victoria
24
Acyclic 3-coloring of triangulated graphs
3
Internal Edge
8
External Edge
6
|E| division vertices
1
7
3
1
1
4
5
3
3
3
1
6/21/2011
2
1
IWOCA 2011, Victoria
25
Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
26
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
27
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
28
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
29
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
30
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
3
1
2
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
31
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
3
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
32
Acyclic 4-coloring of triangulated graphs
3
6
1
8
7
3
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
33
Acyclic 4-coloring of triangulated graphs
3
Number of division
vertices is |E| - n
6
1
8
7
3
1
1
4
5
2
3
3
3
1
6/21/2011
1
2
IWOCA 2011, Victoria
2
34
Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
35
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7
[Angelini & Frati, 2010] Acyclic three coloring of a planar graph
with degree at most 4 is NP-complete
3
Each of the blue vertices are of
degree is 6
1
3
Infinite number of nodes with
the same color at regular
intervals
1
2
3
2
1
1
6/21/2011
1
3
2
3
1
2
1
3
IWOCA 2011, Victoria
2
3
1
2
1
…
36
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7
How to color?
3
1
2
3
2
3
1
1
Acyclic three coloring of a
graph with degree at most
4 is NP-complete
1
2
3
2
G/
2
1
A graph G with
maximum degree four
Maximum degree of G/ is 7
An acyclic four coloring of G/ must ensure acyclic three coloring in G.
Summary of Our Results
3-connected plane
cubic graph with n
vertices
One subdivision per edge,
Acyclically 3-colorable
At most n/2
division vertices.
Partial k-tree, k ≤ 8
One subdivision per edge,
Acyclically 3-colorable
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
Acyclically 3-colorable, simpler
proof, originally proved by
Angelini & Frati, 2010
Each edge has
exactly one division
vertex
Triangulated plane
graph with n vertices
One subdivision per edge,
Acyclically 4-colorable
At most 2n − 6
division vertices.
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.
6/21/2011
IWOCA 2011, Victoria
38
Open Problems
What is the complexity of acyclic 4-colorings for graphs with maximum
degree less than 7?
What is the minimum positive constant c, such that every triangulated plane
graph with n vertices admits a subdivision with at most cn division vertices that
is acyclically k-colorable, k ∈ {3,4}?
6/21/2011
IWOCA 2011, Victoria
39
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IWOCA 2011, Victoria
40
3
1
3
1
2
3
1
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2
IWOCA 2011, Victoria
41