1. Acyclic List Edge Coloring of Graphs
Ko-Wei Lih
李國偉
Institute of Mathematics
Academia Sinica
A Joint Work with Hsin-Hao Lai（賴欣豪）
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.

4. 5-edge coloring

5. acyclic 5-edge coloring

6. 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).

7. Vizing’s Theorem (1964)
(G)  (G)  (G) + 1
(G): the maximum degree of G
Question: (G)  a(G)  (G) + 1 ?
No!
a(K2n) > (K2n) + 1 = 2n for n  2.

8. Acyclic Edge Coloring Conjecture:
a(G)  (G) + 2
Proposed independently by
Fiamčík in 1978 and
Alon, Sudakov, Zaks in 2001.

9. Fiamčík (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).

10. 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)

11. 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.

12. 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.

13. Nĕsetříl, 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.

14. Fiedorowica, Hałusaczak, 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.

15. 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.

16. Hou, Wu, Liu, Liu (2009):
Let G be a planar graph.
(i) a(G)  max{2(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.

17. Let G be an outerplanar graph with (G)  3. Then
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.

18. 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.

19. Hou, Wu, Liu, Liu (2009): (continued)
Let G be an outerplanar graph with (G)  3. Then
(iii) If (G)  5, then a(G) = (G).

20. 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 K2n+1 is a decomposition of the edges of K2n+1 into 2n + 1 matchings each having n edges such that the union of any two matchings forms a Hamiltonian path.

21. Kotzig’s Conjecture (1963): For any n  2, K2n has a perfect 1-factorization.
Proposition. The following statements are equivalent:
K2n+2 has a perfect 1-factorization.
2. K2n+1 has a perfect near-1-factorization.
3. a(K2n+1) = 2n + 1.

22. Kotzig’s 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.
Kotzig’s 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 > n, then a(G)  (G) + 2.

24. 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.

25. 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.

26. 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.

27. 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).

28. Let e = uv be an edge of the graph G
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) e u v
N1(e)

29. i(e) = max{deg(x) : x  Ni(e)}
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) = max{deg(x) : x  Ni(e)}
for each edge e.

30. 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) = max{alist(G1), alist(G2), , alist(Gk)}.

31. then alist(G) = max{alist(G  u), (G)}.
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) = max{alist(G  u), (G)}.
If G is a tree, then G is acyclically 0-edge choosable and alist(G) = (G). u

32. 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.

33. 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

34. 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.

35. Some conditions:
(i) max{deg(u), deg(w), deg(y)}  3;
(ii) deg(u)  deg(y);
(iii) max{deg(u), deg(y)}  deg(w).

36. 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.

37. Outerplanar graphs

38. Outerplanar graphs
Theorem.
If G is an outerplanar graph, then G is acyclically (0 + 1)-edge choosable and alist(G)  (G) + 1.

39. 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.

40. 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.

41. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)
2＋1＋1＋1＋2＋3＋1＋1＋4＋1＋2
=19 vertices

42. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

43. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

44. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

45. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

46. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

47. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

48. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

49. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

50. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

51. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

52. Attaching a cycle of type (2,1,1,1,2,3,1,1,4,1,2)

53. 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).

54. 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

55. 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. H G

56. 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*

57. 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.

58. 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.

59. 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.

60. Planar graphs
Theorem.
If G is a planar graph, then G is acyclically max{20  2, 1 + 22}-edge choosable.
 alist(G)  max{2(G)  2, (G) + 22}

61. 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;

62. 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.

63. 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

64. 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

65. 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;

66. 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.

67. 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.

68. 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;

69. 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;

70. 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.

71. 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.

72. 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;

73. Planar graphs
Lemma. (continued)
(iii) G contains a vertex of degree d adjacent to d  1 vertices of degree 2, where d  4.

74. 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.

75. 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.

76. 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.

77. List Coloring Conjecture:
For any graph G, list(G) = (G).
Open problem 1:
Does alist(G) = a(G) hold for any graph G?

78. 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?

79. Thank you for your attention.