acyclic list edge coloring of graphs

Acyclic List Edge Coloring of Graphs

Ko-Wei Lih李國偉

Institute of MathematicsAcademia Sinica

A Joint Work with Hsin-Hao Lai（賴欣豪）

NTU Math MonthJuly 7, 2009

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.

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.

5-edge coloring

acyclic 5-edge coloring

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).

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.

Acyclic Edge Coloring Conjecture:

a(G) (G) + 2

Proposed independently by

Fiamčík in 1978 and

Alon, Sudakov, Zaks in 2001.

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).

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)

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.

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.

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.

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.

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.

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.

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.

Hou, Wu, Liu, Liu (2009): (continued)


(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.

Hou, Wu, Liu, Liu (2009): (continued)


(iii) If (G) 5, then a(G) = (G).

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.

Kotzig’s Conjecture (1963): For any n 2, K2n has a perfect 1-factorization.

Proposition. The following statements are equivalent:

1. K2n+2 has a perfect 1-factorization.

2. K2n+1 has a perfect near-1-factorization.

3. a(K2n+1) = 2n + 1.

Kotzig’s Conjecture is known to hold for the following cases:

1. 2n ‒ 1 is a prime.

2. n is a prime.

3. 16 particular values of n.

Kotzig’s Conjecture implies a(K2n) = 2n + 1.

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.

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.

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.

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.

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).

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.

eu v

eu v

N0(e)

N1(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.

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)}.

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

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.

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

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.

Some conditions:

(i) max{deg(u), deg(w), deg(y)} 3;

(ii) deg(u) deg(y);

(iii) max{deg(u), deg(y)} deg(w).

Outerplanar graphs

Let G be an outerplanar graph. Then one of the following holds.

(i) there exists a leaf w;

(ii) 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.

Outerplanar graphs

Outerplanar graphs

Theorem.If G is an outerplanar graph, then G is acyclically (0 + 1)-edge choosable and alist(G) (G) + 1.

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.

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.

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

v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

2 ＋ 1 ＋ 1 ＋ 1 ＋ 2 ＋ 3 ＋ 1 ＋ 1 ＋ 4 ＋ 1 ＋ 2=19 vertices

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

v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3

v4


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3 v5

v4


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3 v5

v4=v6


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3 v5

v4=v6

v9


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3 v5

v4=v6

v9v10


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2v6,1

v6,2

v6,3

v7,1

v8,1v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3 v5

v4=v6

v9v10v11

S(G)


v1,1

v1,2 v2,1 v3,1 v4,1 v5,1 v5,2

v6,1

v6,2

v6,3

v7,1

v8,1

v9,1

v9,2

v9,3v9,4

v10,1v11,1

v11,2

v1

v3 v5

v4=v6

v9v10v11

S (G )

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).

41

k

i il

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

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

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*

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.

41

k

i il

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.

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.

Planar graphs

Theorem. If G is a planar graph, then G is acyclically max{20 2, 1 + 22}-edge choosable.

alist(G) max{2(G) 2, (G) + 22}

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;

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.

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

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

Planar graphs


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;

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.

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.

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;

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;

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.

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.

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;

Planar graphs


adjacent to d 1 vertices of degree 2, where d 4.

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.

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.

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.

List Coloring Conjecture:

For any graph G, list(G) = (G).

Open problem 1:

Does alist(G) = a(G) hold for any graph G?

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?

Thank you for your attention.

acyclic list edge coloring of graphs

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