Acyclic List Edge Coloring of Graphs

1
Acyclic List Edge Coloring of Graphs

• Ko-Wei Lih
• ???
• Institute of Mathematics
• Academia Sinica
• A Joint Work with Hsin-Hao Lai(???)
• NTU Math Month
• July 7, 2009

2
All graphs in this talk are finite, without loops
or parallel edges. The chromatic number ?(G) of
G is the least number of colors in a proper
vertex coloring of G. The chromatic index ??(G)
of G is the least number of colors in a proper
edge coloring of G.
3
A proper coloring of the vertices or edges of a
graph G is called acyclic if there is no
2-colored cycle in G. ? Every cycle of G is
colored with at least 3 colors. ? The union of
any two color classes induces a subgraph of G
which is a forest.
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5-edge coloring
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acyclic 5-edge coloring
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The acyclic chromatic number a(G) of G is the
least number of colors in an acyclic vertex
coloring of G. There has been a large number of
works on a(G). The acyclic chromatic index a?(G)
of G is the least number of colors in an acyclic
edge coloring of G. Lesser is known about a?(G).
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Vizings Theorem (1964) ?(G) ? ??(G) ? ?(G)
1 ?(G) the maximum degree of G Question ?(G)
? a?(G) ? ?(G) 1 ? No! a?(K2n) gt ?(K2n) 1
2n for n ? 2.
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Acyclic Edge Coloring Conjecture a?(G) ? ?(G)
2 Proposed independently by Fiamcík in 1978
and Alon, Sudakov, Zaks in 2001.
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Fiamcík (1984) If ?(G) ? 3 and no component of G
is K4 or K3,3, then a?(G) ? 4, whereas a?(K4)
a?(K3,3) 5. Alon, Sudakov, Zaks (2001) There
exists a constant c such that a?(G) ? ?(G) 2
for any G whose girth, the length of a shortest
cycle, of G is at least c?(G)log?(G).
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Molloy, Reed (1998) a?(G) ? 16?(G) Muthu,
Narayanan, Subramanian (2005) When the girth of
G is at least 220, a?(G) ? 4.52?(G)
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Muthu, Narayanan, Subramaniann (2005) a?(G) ?
?(G) 1 if G is a partial 2-tree, an outerplanar
graph, or a partial torus. Basavaraju, Sunil
Chandran (2008) a?(G) ? ?(G) 1 if G is a
2-degenerate graph.
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Basavaraju, Sunil Chandran (2009) a?(G) ? 6 if G
is connected, ?(G) ? 4 and m ? 2n ? 1, where m is
the number of edges of G and n is the number of
vertices of G. In general, a?(G) ? 7 if ?(G) ? 4.
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Nesetríl, Wormald (2005) a?(G) ? ?(G) 1 for a
random ?regular graph. Skulrattankulchai (2004)
A polynomial time algorithm to color a subcubic
graph using 5 colors. Alon, Zaks (2002) It is
NP-complete to determine whether a?(G) ? 3.
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Fiedorowica, Halusaczak, Narayanan (2008) a?(G)
? ?(G) 6 if G is a planar graph without
3-cycles or G has an edge-partition into two
forests. a?(G) ? 2?(G) 29 if G is a planar
graph.
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Borowiecki, Fiedorowicz (2009) a?(G) ? ?(G) 2
for any planar graph G if the girth of G is at
least 5 or G contains no cycles of length 4, 6,
8, 9. a?(G) ? ?(G) 1 for any planar graph G of
girth at least 6. a?(G) ? ?(G) 15 for any
planar graph G without 4-cycles.
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Hou, Wu, Liu, Liu (2009) Let G be a planar
graph. (i) a?(G) ? max2?(G) ? 2, ?(G) 22
when girth(G) ? 3. (ii) a?(G) ? ?(G) 2 when
girth(G) ? 5. (iii) a?(G) ? ?(G) 1 when
girth(G) ? 7. (iv) a?(G) ?(G) when girth(G) ?
16 and ?(G) ? 3.
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Hou, Wu, Liu, Liu (2009) Let G be an outerplanar
graph with ?(G) ? 3. Then (i) If ?(G) 3, then
a?(G) 4 if G contains a subgraph isomorphic to
the graph Pm. Otherwise a?(G) 3.
Vertices marked have no edges of G incident
with them other than those shown and pair of
vertices marked with ? can be connected to each
other.
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Hou, Wu, Liu, Liu (2009) (continued) Let G be an
outerplanar graph with ?(G) ? 3. Then (ii) If
?(G) 4, then a?(G) 5 if G contains a subgraph
isomorphic to the graph Q. Otherwise a?(G) 4.
Vertices marked have no edges of G incident
with them other than those shown and pair of
vertices marked with ? can be connected to each
other.
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Hou, Wu, Liu, Liu (2009) (continued) Let G be an
outerplanar graph with ?(G) ? 3. Then (iii) If
?(G) ? 5, then a?(G) ?(G).
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A perfect 1-factorization of K2n is a
decomposition of the edges of K2n into 2n ? 1
perfect matchings such that the union of any two
matchings forms a Hamiltonian cycle. A perfect
near-1-factorization of K2n1 is a decomposition
of the edges of K2n1 into 2n 1 matchings each
having n edges such that the union of any two
matchings forms a Hamiltonian path.
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• Kotzigs Conjecture (1963) For any n ? 2, K2n
  has a perfect 1-factorization.
• Proposition. The following statements are
  equivalent
• K2n2 has a perfect 1-factorization.
• 2. K2n1 has a perfect near-1-factorization.
• 3. a?(K2n1) 2n 1.

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• Kotzigs Conjecture is known to hold for the
  following cases
• 2n ? 1 is a prime.
• 2. n is a prime.
• 3. 16 particular values of n.
• Kotzigs Conjecture implies a?(K2n) 2n 1.

23
Alon, Sudakov, Zaks (2001) suggested a
possibility that complete graphs of even order
are the only regular graphs which require ? 2
colors to be acyclically edge colored. Basavaraju,
Sunil Chandran, Kummini (2009) Let G be a
d-regular graph with 2n vertices and d gt n, then
a?(G) ? ?(G) 2.
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Basavaraju, Sunil Chandran, Kummini (2009) For
any d and n such that dn is even and d ? 5, n ?
2d 3, then there exists a connected d-regular
graph with n vertices that requires d 2 colors
to be acyclically edge colored. a?(Kn,n) ? n 2
?(Kn,n) 2, when n is odd.
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Basavaraju, Sunil Chandran (2009)
(continued) a?(Kp,p) p 2 ?(Kp,p) 2,
when p is an odd prime. If G is obtained from
Kp,p by removing an edge, then a?(G) ? ?(G) 1.
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An edge-list L assigns a finite set of positive
integers to each edge of G. Let f E(G) ? N. An
edge-list L is an f-edge-list if L(e) f(e)
for every edge e. An acyclic edge coloring ? of G
such that ?(e) ? L(e) for every edge e is called
an acyclic L-edge coloring of G.
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A graph G is said to be acyclically f-edge
choosable if it has an acyclic L-edge coloring
for any f-edge-list L. The acyclic list
chromatic index alist?(G) is the least integer k
such that G is acyclically k-edge choosable.
Obviously, ?(G) ? ??(G) ? a?(G) ? alist?(G).
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Let e uv be an edge of the graph G. Let N0(e)
and N1(e) denote the sets u, v and x xu ?
E(G) or xv ? E(G), respectively.
e
u
v
N0(e)
N1(e)
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Let e uv be an edge of the graph G. Let N0(e)
and N1(e) denote the sets u, v and x xu ?
E(G) or xv ? E(G), respectively. For i 0 and
1, let ?i denote the mapping ?i(e) maxdeg(x)
x ? Ni(e) for each edge e.
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Lemma. Assume that f1 and f2 are two mappings
from E(G) to N such that f1(e) ? f2(e) for each
e. If G is acyclically f1-edge choosable, then G
is acyclically f2-edge choosable. Lemma. If H is
a subgraph of a graph G, then alist?(H) ?
alist?(G). Lemma. If G1, G2, . . . , Gk are all
the components of G, then alist?(G)
maxalist?(G1), alist?(G2), . . . , alist?(Gk).
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Adding a leaf Let u be a leaf of G. If G ? u is
acyclically ?0-edge choosable, so is G. If u is a
leaf of G, then alist?(G) maxalist?(G ? u),
?(G). If G is a tree, then G is acyclically
?0-edge choosable and alist?(G) ?(G).
u
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Adding a vertex of degree 2 Let w be a vertex of
degree 2 in G. Let P uvwx be a path of G such
that (i) vx ? E(G) (ii) deg(v) ? 3 (iii) deg(u)
? 2 when deg(v) 3. If G ? w is acyclically (?0
1)-edge choosable, so is G.
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Subdividing an edge If G is obtained from an
acyclically (?1 1)-edge choosable graph H by
subdividing an edge, then G is acyclically (?1
1)-edge choosable.
H
G
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Joining two vertices of degree 2 If G is
obtained from an acyclically (?1 1)-edge
choosable graph H by adding an edge between two
vertices of degree 2 with a unique common
neighbor (under some conditions), then G is
acyclically (?1 1)-edge choosable.
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Some conditions (i) maxdeg(u), deg(w),
deg(y) ? 3 (ii) deg(u) ? deg(y) (iii)
maxdeg(u), deg(y) ? deg(w).
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• Outerplanar graphs
• Let G be an outerplanar graph. Then one of the
  following holds.
• (i) there exists a leaf w
• there exists an edge vw such that deg(v) ? 3 and
  deg(w) 2
• (iii) there exists edges uv and vw such that
  deg(u) 2, deg(v) 4, and deg(w) 2.

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Outerplanar graphs
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Outerplanar graphs Theorem. If G is an
outerplanar graph, then G is acyclically (?0
1)-edge choosable and alist?(G) ? ?(G) 1.
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Non-regular subcubic graphs Theorem. If G
satisfies ?(G) ? 3 and ?(G) ? 2, then G is
acyclically (?0 1)-edge choosable and alist?(G)
? ?(G) 1.
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Cubic graphs with triangles Theorem. If G is a
cubic graph, G contains a triangle, and G ? K4,
then G is acyclically (?0 1)-edge choosable and
alist?(G) ? ?(G) 1.
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Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)
21112311412 19 vertices
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle of type(2,1,1,1,2,3,1,1,4,1,2)
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Attaching a cycle Lemma. Assume that S(G) is a
graph obtained from G by attaching a cycle of
type (l1,l2,,lk), where . Let li
? 2 for some i or let vj,1 have no neighbor in G
for some j. If G is acyclically (?1 1)-edge
choosable, so is S(G).
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• Halin Graphs
• A Halin graph H is a plane graph obtained by
  drawing a tree Tr in the plane, where Tr has no
  vertex of degree 2, and a cycle C through all
  leaves of Tr in the plane.

Tr
C
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• Subdivisions
• A graph G is called a subdivision of a graph H if
  G can be obtained from H by inserting new
  vertices in edges of H.

G
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• Subdivisions of Halin graphs
• Theorem.If G Tr ? C is a subdivision of a
  Halin graph H Tr ? C and G ? K4, then G is
  acyclically (?1 1)-edge choosable and alist?(G)
  ? ?(G) 1.

Tr
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Attaching a cycle Lemma. Assume that S(G) is a
graph obtained from G by attaching a cycle of
type (l1,l2,,lk), where . Let li
? 3 for some i and vj ? 2 for each j. If G is
acyclically ?0-edge choosable, then S(G) is
acyclically max?0, 6-edge choosable.
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Halin graphs Theorem. If H Tr ? C is Halin
graph that contains two 3-faces sharing a common
edge, then H is acyclically max?0, 6-edge
choosable. In particular, alist?(H) ?(H) when
?(H) ? 6.
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Planar graphs Lemma. Let G be a planar graph.
Then there exists a vertex v with k neighbors v1,
v2, . . . , vk (deg(v1) ? . . . ? deg(vk)) such
that one of the following holds (i) k ?
2 (ii) k 3 with deg(v1) ? 11 (iii) k 4
with deg(v1) ? 7, deg(v2) ? 11 (iv) k 5 with
deg(v1) ? 6, deg(v2) ? 7, deg(v3) ? 11.
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Planar graphs Theorem. If G is a planar graph,
then G is acyclically max2?0 ? 2, ?1 22-edge
choosable. ? alist?(G) ? max2?(G) ? 2, ?(G)
22
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Planar graphs Lemma. Let G be a planar graph
with ?(G) ? 2. If any two 4-cycles are
vertex-disjoint and there is no 3-cycle, then one
of the following holds (i) G contains an edge
with one endpoint of degree 2 and the other
endpoint of degree at most 4 (ii) G contains a
vertex of degree 3 adjacent to two vertices of
degree 3
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Planar graphs Lemma. (continued) (iii) G
contains a vertex of degree d adjacent to d ? 3
vertices of degree 2, where d ? 5 (iv) G
contains a vertex of degree 4 adjacent to three
vertices of degree 3 (v) G contains a face f
v1v2v3v4 with deg(v1) 2 and deg(v2) 5.
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Planar graphs Theorem. If G is a planar graph
such that any two 4-cycles are vertex-disjoint
and there is no 3-cycle, then G is acyclically
(?1 3)-edge choosable. ? alist?(G) ? ?(G) 3
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Planar graphs Lemma. Let G be a planar graph
with ?(G) ? 2 and girth(G) ? 5, then one of the
following holds (i) G contains a vertex of
degree 2 adjacent to a vertex of degree at most
3 (ii) G contains a vertex of degree 3 adjacent
to two vertices of degree 3
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Planar graphs Lemma. (continued) (iii) G
contains a vertex of degree d adjacent to d ? 2
vertices of degree 2, where d ? 4 (iv) G
contains a vertex of degree 4 adjacent to a
vertex of degree 2 and a vertex of degree 3
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Planar graphs Lemma. (continued) (v) G contains
a vertex of degree 5 adjacent to two vertices of
degree 2 and a vertex of degree 3 (vi) G
contains a face f v1v2v3v4v5 with deg(v1)
deg(v4) 2, deg(v2) deg(v3) 4 and deg(v5)
5.
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Planar graphs Theorem. If G is a planar graph
with girth(G) ? 5, then G is acycically max?0
2, 6-edge choosable. ? alist?(G) ? ?(G) 2
when ?(G) ? 4.
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Planar graphs Lemma. Let G be a planar graph
with ?(G) ? 2. If any two 4-cycles are
edge-disjoint and there are neither 3-cycles nor
5-cycles, then one of the following holds (i) G
contains an edge with one endpoint of degree 2
and the other endpoint of degree at most 3
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Planar graphs Lemma. (continued) (ii) G
contains a vertex of degree 3 adjacent to two
vertices of degree 3 (iii) G contains a vertex
of degree d adjacent to d 2 vertices of degree
2, where d ? 4
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Planar graphs Lemma. (continued) (iv) G contains
a vertex of degree 4 adjacent to a vertex of
degree 2 and a vertex of degree 3 (v) G contains
a face f v1v2v3v4 with deg(v1) 2 and deg(v2)
4.
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Planar graphs Theorem. If G is a planar graph
such that any two 4-cycles are edge-disjoint and
there are neither 3-cycles nor 5-cycles, then G
is acyclically max?1 2, 6-edge choosable. ?
alist?(G) ? ?(G) 2 when ?(G) ? 4.
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Planar graphs Lemma. Let G be a planar graph
with ?(G) ? 2 and girth(G) ? 7. Then one of the
following holds (i) G contains a vertex of
degree 2 adjacent to a vertex of degree 2 (ii) G
contains a vertex of degree 3 adjacent to a
vertex of degree 2 and a vertex of degree at most
3
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Planar graphs Lemma. (continued) (iii) G
contains a vertex of degree d adjacent to d ? 1
vertices of degree 2, where d ? 4.
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Planar graphs Theorem. If G is a planar graph
with girth(G) ? 7, then G is acyclically (?1
1)-edge choosable. ? alist?(G) ? ?(G) 1.
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Planar graphs Lemma. Let G be a planar graph
with ?(G) ? 2. If girth(G) ? 16, then G has a
vertex of degree 2 whose neighbors are also of
degree 2.
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Planar graphs Theorem. If G is a planar graph
with girth(G) ? 16, then G is acyclically max?0,
3-edge choosable. ? alist?(G) ? ?(G) if ?(G) ?
3.
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List Coloring Conjecture For any graph G,
?list?(G) ??(G). Open problem 1 Does
alist?(G) a?(G) hold for any graph G?
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Open problem 2 Does alist?(G) ? ?(G) 2 hold
for any graph G? Stronger forms Is G
acyclically (?0 2)-edge choosable for any
G? Is G acyclically (?1 2)-edge choosable for
any G?
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Thank you for your attention.