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

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.

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.

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.

Hou, Wu, Liu, Liu (2009): (continued) Let G be an outerplanar graph with (G)  3. Then (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: 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: 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 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)

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.

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)}.

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

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; 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) 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)

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

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

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

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

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

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

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

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

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

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

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).

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.

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{20  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 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;

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 Lemma. (continued) (iii) G contains a vertex of degree d 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.