Download - 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)
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:
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
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
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
v3
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
v3
v4
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
v3 v5
v4
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
v3 v5
v4=v6
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
v3 v5
v4=v6
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
v3 v5
v4=v6
v9
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
v3 v5
v4=v6
v9v10
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 v5,2v6,1
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)
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
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
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.