[PPT] - Adjacent Vertex Distinguishing Definitions Some Known Results PowerPoint Presentation

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Adjacent Vertex Distinguishing Colorings of Graphs

Wang Weifan Department of Mathematics Zhejiang Normal University Jinhua 321004

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Definitions
Some Known Results
Our Main Results
Outline of Proofs

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1

Definitions

¶ Let G = (V, E) be a simple graph. If G is a plane

graph, let F denote the set of faces in G.

¶ Edge-k-coloring:

A mapping f : E → {1, 2, . . . , k} such that f(e) = f(e′) for any adjacent edges e, e′ ∈ E.

¶ Edge chromatic number:

χ′(G) = min{k|G is edge-k-colorable}.

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¶ Total-k-coloring:

A mapping f : V ∪ E → {1, 2, . . . , k} such that any two adjacent vertices, adjacent edges, and incident vertex and edge are assigned to different colors.

¶ Total chromatic number:

χ′′(G) = min{k|G is total-k-colorable}.

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¶ For an edge coloring f of G and for a vertex

v ∈ V , we define: Cf(v) = {f(e)|e is incident to v}.

¶ For a total coloring f of G and for a vertex

v ∈ V , we define: Cf[v] = {f(e)|e is incident to v} ∪ {f(v)}.

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¶ Vertex-Distinguishing edge coloring (VD edge

coloring) or Strong edge coloring: A proper edge coloring f such that Cf(u) = Cf(v) for any two vertices u, v ∈ V .

¶ Vertex-Distinguishing edge chromatic number

(VD edge chromatic number): χ′

s(G) = min{k|G is VD edge-k-colorable}.

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¶ Vertex-Distinguishing total coloring (VD total

coloring) or Strong total coloring:: A proper total coloring f such that Cf[u] = Cf[v] for any two vertices u, v ∈ V .

¶ Vertex-Distinguishing total chromatic number

(VD edge chromatic number): χ′′

s(G) = min{k|G is VD total-k-colorable}.

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¶ Adjacent-Vertex-Distinguishing edge coloring

(AVD edge coloring): A proper edge coloring f such that Cf(u) = Cf(v) for any adjacent vertices u, v ∈ V .

¶ Adjacent-Vertex-Distinguishing edge chromatic

number (AVD edge chromatic number): χ′

a(G) = min{k|G is AVD edge-k-colorable}.

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¶ Adjacent-Vertex-Distinguishing total coloring

(AVD total coloring): A proper total coloring f such that Cf[u] = Cf[v] for any adjacent vertices u, v ∈ V .

¶ Adjacent-Vertex-Distinguishing total chromatic

number (AVD edge total chromatic number): χ′′

a(G) = min{k|G is AVD total-k-colorable}.

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Examples First Example: C5

χ′(C5) = 3, χ′

a(C5) = 5,

χ′′

a(C5) = χ′′(C5) = 4.

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1 2 5 4 3 3 2 1 4 1 1 4 4 2 3

' 5

( ) 5

a C

χ =

" 5

( ) 4

a C

χ =

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Second Example: K4

χ′(K4) = 3, χ′

a(K4) = 5,

χ′′

a(K4) = χ′′(K4) = 5.

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3 1 4 1 5 2 2 4 5 3 1 5 1 4 3 2

" 4

( ) 5

a K

χ =

' 4

( ) 5

a K

χ =

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2

Some Known Results ∆: the maximum degree of a graph G δ: the minimum degree of a graph G d-Vertex: a vertex of degree d n: the number of vertices of a graph G nd: the number of d-vertices in G

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(2.1) Strong Edge Coloring (VD Edge Coloring) The concept of strong edge coloring was intro- duced independently by Aigner, Triesch, and Tuza, by Horn´ ak and Sot´ ak, and by Burris and Schelp.

A graph G has a strong edge coloring if and only if G contains no isolated edges, and G has at most one isolated vertex. In this part, we assume that G has no isolated edges and has at most one isolated vertex.

Combinatorial degree µ(G): µ(G) = max

δ≤d≤∆ min{k|

k

d

≥ nd}.

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Conjecture 2.1 (Burris and Schelp, 1997) For a graph G, µ(G) ≤ χ′

s(G) ≤ µ(G) + 1.

Conjecture 2.2 (Burris and Schelp, 1997) For a graph G, χ′

s(G) ≤ n + 1.

[A.C.Burris, R.H.Schelp, J.Graph Theory, 26(1997) 73-82.]

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Theorem 2.1.1 For n ≥ 3, χ′

s(Kn) = n if n is odd,

and χ′

s(Kn) = n + 1 if n is even.

Theorem 2.1.2 Let n be a cycle of length n ≥ 3 and let µ = µ(Cn). Then χ′

s(Cn) = µ + 1 if µ is

dd and

µ

2

− 2 ≤ n ≤

µ

2

− 1 or µ is even and

n > (µ2 − 2µ)/2, and χ′

s(Cn) = µ otherwise.

[A.C.Burris and R.H.Schelp, J.Graph Theory, 26(1997) 73-82.]

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Theorem 2.1.3 For a graph G, m1 ≤ χ′

s(G) ≤ (∆+

1)⌊2m2 + 5⌋, where m1 = max

1≤k≤∆{(k!nk)

1 k + k−1

2 },

m2 = max

1≤k≤∆ n

1 k

k.

Corollary 2.1.4 If G is an r-regular graph of order n, then χ′

s(G) ≤ (r + 1)⌊2n

1 r + 5⌋. [A.C.Burris,R.H.Schelp, J.Graph Theory, 26(1997) 73-82.]

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Let c be the smallest number in the interval (4, 6.35) such that

6c2+c(49c2−208)

1 2

c2−16

< max{n1 + 1, ⌈cn

1 2

2⌉, 21}.

Theorem 2.1.5 For a tree T = K2, we have χ′

s(T) ≤ max{n1 + 1, ⌈cn

1 2

2⌉, 21}.

[A.C.Burris, R.H.Schelp, J.Graph Theory, 26(1997) 73-82.]

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Theorem 2.1.6 Let G be a vertex-disjoint union of cycles, and let k be the least number such that n ≤ k

2

. Then χ′

s(G) = k or k + 1.

Theorem 2.1.7 Let G be a vertex-disjoint union of paths with each path of length at least two. Let k be the least number such that n1 ≤ k and n2 ≤ k

2

.

Then χ′

s(G) = k or k + 1.

Theorem 2.1.8 Let G be a strong edge colorable graph with ∆ = 2. Let k be the least number such that n1 ≤ k and n2 ≤ k

2

. Then k ≤ χ′

s(G) ≤

k + 5.

[P.N.Balister, B.Bollb´ as, R.H.Schelp, Discrete Math., 252(2002) 17-29.]

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Theorem 2.1.9 If Lm is an m-sided prism, then χ′

s(Lm) ≤ µ(Lm) + 1.

Let G be a graph and r ≥ 1 be an integer. Let rG denote the graph obtained from G by replacing each edge of G with r multi-edges. Theorem 2.1.10 Let r ≥ 1 be an integer. Then χ′

s(rK4) ≤ µ(rK4) + 1.

[K.Taczuk, M.Wo´ zniak, Opuscula Math., 24/2(2004) 223-229.]

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Theorem 2.1.11 Let G be disjoint union of suffi- ciently many k-regular 1-factorizable graphs. Then χ′

s(G) ≤ µ(G) + 1.

Theorem 2.1.12 Let G be 3-regular graph with 1- factor on at most 12 vertices. Then, for each posi- tive integer r, χ′

s(G) ≤ µ(rG) + 1.

Theorem 2.1.13 Let G ∈ {K4,4, K5,5, K6,6, K7,7, K6}. Then, for each integer r, χ′

s(G) ≤ µ(rG) + 1.

[J.Rudaˇ sov´ a, R.Sot´ ak, Discrete Math., 308(2008) 795-802.]

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A proper edge k-coloring of a graph G is called equitable if the number

f edges in any two color classes differ by at most one.

It is well known that if G is edge k-colorable, then G also is equitably edge k-colorable.

Theorem 2.1.14 For any integer k ≥ χ′

s(G), G has

an equitable strong edge k-coloring.

[J.Rudaˇ sov´ a, R.Sot´ ak, Discrete Math., 308(2008) 795-802.]

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For a vertex v ∈ V (G), define a split at v to be a new graph G′ in which v is replaced by two nonadjacent vertices v1 and v2 with the neighborhood

f v in G equal to the disjoint union of the neighborhoods of v1 and v2 in

G′. Call a split G′ an r-split if the degree of v1 (or v2) is r.

Theorem 2.1.15 Let G be a graph with n∆ = 1, and let k ≥ ∆. If there is a 2-split G′ of G at v with χ′

s(G′) ≤ k − 1, then χ′ s(G) ≤ k.

[P.B.Balister, Random Struct. Alg., 20(2001) 89-97.]

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Theorem 2.1.16 If n∆ = 1, n2 = n∆−1 = 0, n0, n1, n∆−1 ≤ 1, n3, n4 ≤ 2, n∆−3 ≤ ∆ − 1, and for 5 ≤ d ≤ ∆ − 4, nd ≤ d − 4 d − 3 min{2 ∆ − 5 ∆ − 3

,

∆ − 2 d

} − 2,

then χ′

s(G) = ∆.

Theorem 2.1.17 If Gn,p is a random graph on n vertices with edge probability p and pn

log n, (1−p)n log n →

∞, then Pro(χ′

s(Gn,p) = ∆) → 1 as n → ∞.

[P.B.Balister, Random Struct. Alg., 20(2001) 89-97.]

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Theorem 2.1.18 Let c be a real number with 1

2 <

c ≤ 1. Let G be a graph on n vertices. If δ ≥ 5 and ∆ < (2c−1)n−4

3

, then χ′

s(G) ≤ ⌈cn⌉.

[O.Favaron, H.Li, R.H.Schelp, Discrete Math., 159(1996) 103-109.]

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Theorem 2.1.19 Let G be a graph on n ≥ 3 ver-

tices. If δ > n

3, then χ′ s(G) ≤ ∆ + 5.

[C.Bazgan, H.Li, M.Wo´ zniak, Discrete Math., 236(2001) 37-42.]

Theorem 2.1.20 Let G be a graph on n vertices. Then χ′

s(G) ≤ n + 1. [Conjecture 2.2 is true.]

[C.Bazgan, A.Harkat-Benhamdine, H.Li, J. Combin. Theory Ser.B., 75 (1999) 288-301.]

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(2.2) AVD Edge Coloring Normal graph: a graph without isolated edges

A graph G has an AVD edge coloring if and only if G contains no isolated edges. Thus, we always assume that G is a normal graph in this subsection.

Theorem 2.2.1 (Vizing, 1964) For a simple graph G, ∆ ≤ χ′(G) ≤ ∆ + 1.

G is Class 1 if χ′(G) = ∆, and Class 2 if χ′(G) = ∆ + 1.

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Conjecture 2.3 (Z.Zhang, L.Liu, J.Wang, 2002) For a normal graph G (= C5), χ′

a(G) ≤ ∆ + 2.

Note: χ′

a(C5) = 5 = ∆ + 3. Is unique C5 as an exception?

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¶ ∆ ≤ χ′(G) ≤ χ′

a(G).

¶ If G has two adjacent ∆-vertices, then χ′

a(G) ≥

∆ + 1.

¶ If any two adjacent vertices of a graph have dis-

tinct degree, then χ′

a(G) = ∆.

¶ For a cycle Cn, χ′

a(Cn) = 5 if n = 5, 3 if n ≡ 0

(mod 3), and 4 otherwise.

[Z.Zhang, L.Liu, J.Wang, Appl. Math. Lett., 15(2002) 623-626]

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¶ For a complete graph Kn with n ≥ 3, χ′

a(Kn) =

n if n ≡ 0 (mod 2), and χ′

a(Kn) = n+1 if n ≡ 1

(mod 2).

¶ For a complete bipartite graph Km,n with 1 ≤

m ≤ n, χ′

a(Km,n)

= n if m < n, and χ′

a(Km,n) = n + 2 if m = n.

¶ For a tree T, χ′

a(T) ≤ ∆ + 1; χ′ a(T) = ∆ + 1 ⇔

T has adjacent ∆-vertices.

[Z.Zhang, L.Liu, J.Wang, Appl. Math. Lett., 15(2002) 623-626]

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Theorem 2.2.2 For any graph G, χ′

a(G) ≤ 3∆.

[S.Akbari, H.Bidkhori, N.Nosrati, Discrete Math. 306(2006) 3005-3010.] [M.Ghandehari, H.Hatami, Two upper bounds for the strong edge chromatic number, preprint.]

Theorem 2.2.3 For any graph G, χ′

a(G) ≤ 3∆ − 1.

[Y.Dai, Y.Bu, Math. Econ., 26(1)(2009) 107-110.]

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Theorem 2.2.4 For a graph G with ∆ ≥ 106, we have χ′

a(G) ≤ ∆ + 27

√ ∆ ln ∆. Theorem 2.2.5 For an ∆-regular graph G, with ∆ > 100, we have χ′

a(G) ≤ ∆ + 3 log2 ∆.

[M.Ghandehari, H.Hatami, Two upper bounds for the strong edge chromatic number, preprint.]

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Theorem 2.2.6 For a graph G with ∆ > 1020, then χ′

a(G) ≤ ∆ + 300.

[H.Hatami, J. Combin. Theory Ser. B, 95(2005) 246-256. ]

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Theorem 2.2.7 If G is a graph with ∆ = 3, then χ′

a(G) ≤ 5.

Theorem 2.2.8 If G is a bipartite graph, then χ′

a(G) ≤ ∆ + 2.

Theorem 2.2.9 For any graph G, χ′

a(G) ≤ ∆ +

O(logχ(G)), where χ(G) is the vertex chromatic number of G.

[P.N.Balister, E.Gy˝

ri, J.Lehel, R.H.Schelp, SIAM J. Discrete Math.

21(2007) 237-250.]

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Theorem 2.2.10 If G is a graph with ∆ = 4, then χ′

a(G) ≤ 8.

[Y.Dai, Master Thesis, 2007]

Theorem 2.2.11 Let r ≥ 4. Then a random r- regular graph G asymptotically almost surely has χ′

a(G) ≤ ⌈3r/2⌉.

Corollary 2.2.12 A random 4-regular graph G asymptotically almost surely has χ′

a(G) ≤ 6 =

∆ + 2.

[C.Greenhill, A.Ruc´ nski, The Electronic J. Combin., 13(2006),♯R77.]

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Theorem 2.2.13 If G is a planar bipartite graph with ∆ ≥ 12, then χ′

a(G) ≤ ∆ + 1.

Corollary 2.2.14 Let G be a planar bipartite graph with ∆ ≥ 12. If G contains two adjacent ∆- vertices, then χ′

a(G) = ∆ + 1.

[K.Edwards, M.Horˇ n´ ak, M.Wo´ zniak, Graphs Combin., 22(2006) 341- 350.]

Question 2.4 Is necessary Corollary 2.2.14? (If such

graph has no adjacent ∆-vertices, then χ′

a(G) = ∆?)

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Theorem 2.2.15 If G is a Hamiltonian graph with χ(G) ≤ 3, then χ′

a(G) ≤ ∆ + 3.

Theorem 2.2.16 If G is a graph with χ(G) ≤ 3 and G has a Hamiltonian path, then χ′

a(G) ≤ ∆ + 4.

A subgraph H of a graph G is called a dominating subgraph of G if V (G) − V (H) is an independent set.

Theorem 2.2.17 If a graph G has a dominat- ing cycle or a dominating path H such that χ(G[V (H)]) ≤ 3, then χ′

a(G) ≤ ∆ + 5.

[B.Liu, G.Liu, Intern. J. Comput. Math., 87(2010) 726-732.]

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The p-dimensional hypercube Qp is the graph whose vertices are the or- dered p-tuples of 0’s and 1’s, two vertices being adjacent if and only if they differ in exactly one coordinate. For example, Q2 is a 4-cycle, and Q3 is the cube.

Theorem 2.2.18 χ′

a(Qp) = p + 1 for all p ≥ 3.

[M.Chen, X.Guo, Inform. Process. Lett., 109(2009) 599-602.]

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Conjecture 2.5 (TCC) [M.Behzad 1965; V,G.Vizing, 1968] For a simple graph G, ∆ + 1 ≤ χ′′(G) ≤ ∆ + 2. Theorem 2.2.19 For any simple graph G, χ′′(G) ≤ ∆ + 1026.

[M.Molloy, B.Reed, Combinatorics, 18(1998), 214-280.]

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Conjecture 2.6 For a graph G with no K2 or C5 component, χ′

a(G) ≤ χ′′(G).

Conjecture 2.7 For a r-regular graph G with no C5 component (r ≥ 2), χ′

a(G) = χ′′(G).

[Z.Zhang, D.R.Woodall, B.Yao, J.Li, X.Chen, L.Bian, Sci. China Ser.A, 52(2009) 973-980.]

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Theorem 2.2.20 Conjecture 2.7 holds for all regu- lar graphs in the following classes:

Regular graphs G with χ′′(G) = ∆ + 1;
2-Regular graphs and 3-regular graphs;
Bipartite regular graphs;
Complete regular multipartite graphs;
Hypercubes;
Join graphs Cn

Cn;

(n − 2)-Regular graphs of order n.

[Z.Zhang, D.R.Woodall, B.Yao, J.Li, X.Chen, L.Bian, Sci. China Ser.A, 52 (2009) 973-980.]

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(2.3) AVD Total Coloring Conjecture 2.8 For a graph G with n ≥ 2 vertices, χ′′

a(G) ≤ ∆ + 3.

∆ + 1 ≤ χ′′(G) ≤ χ′′

a(G).

If G has two adjacent ∆-vertices, then χ′′

a(G) ≥

∆ + 2.

[Z.Zhang, X.Chen, J.Li, B.Yao, X.Lu, J.Wang, Sci. China Ser. A 34 (2004) 574-583]

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¶ If n ≥ 4, then χ′′

a(Cn) = 4.

¶ χ′′

a(Kn) = n + 1 if n is even, χ′′ a(Kn) = n + 2

therwise.

¶ Let n + m ≥ 2. Then χ′′

a(Km,n) = ∆ + 1 if

m = n, χ′′

a(Km,n) = ∆ + 2 if m = n.

¶ For a tree T with n ≥ 2 vertices, χ′′

a(T) ≤ ∆+2;

χ′′

a(T) = ∆ + 2 ⇔ T has adjacent ∆-vertices.

[Z.Zhang, X.Chen, J.Li, B.Yao, X.Lu, J.Wang, Sci. China Ser. A 34 (2004) 574-583]

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Theorem 2.3.1 For a graph G with n ≥ 2 vertices, χ′′

a(G) ≤ χ(G) + χ′(G).

¶ If G is a bipartite graph, then χ′′

a(G) ≤ ∆ + 2;

moreover, χ′′

a(G) = ∆ + 2 if G contains adjacent

∆-vertices.

¶ If G is planar, then χ′′

a(G) ≤ 4 + ∆ + 1 = ∆ + 5.

When G is Class 1, χ′′

a(G) ≤ 4 + ∆ = ∆ + 4.

(Using Four-Color Theorem and Vizing Theorem) ¶ If G is Class 1 and χ(G) ≤ 3, then χ′′

a(G) ≤

∆ + 3.

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Theorem 2.3.2 If G is a graph with ∆ = 3, then χ′′

a(G) ≤ 6.

[X.Chen, Discrete Math. 308(2008) 4003-4008.] [H.Wang, J. Comb. Optim. 14(2007) 87-109.] [J.Hulgan, Discrete Math. 309(2009) 2548-2550.]

Question 2.9 [J.Hulgan, 2009] For a graph G with ∆ = 3, is the bound χ′′

a(G) ≤ 6 sharp?

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Theorem 2.3.3 Let Qp be a p-dimensional hyper- cube with p ≥ 2, then χ′′

a(Qp) = p + 2.

[M.Chen, X.Guo, Inform. Process. Lett., 109(2009) 599-602.] A connected graph G is called a 1-tree if there is a vertex v ∈ V (G) such that G − v is a tree.

Theorem 2.3.4 If G is a 1-tree, then ∆ + 1 ≤ χ′′

a(G) ≤ ∆ + 2; and χ′′ a(G) = ∆ + 2 if and only if

G contains two adjacent ∆-vertices.

[H.Wang, Ars Combin., 91(2009) 183-192].

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3

Our Main Results (3.1) χ′′

a for outerplanar graphs

A planar graph is called outerplanar if there is an embedding of G into the Euclidean plane such that all the vertices are incident to the unbounded face. Note that if G is an outerplanar graph with ∆ ≥ 3, then χ(G) ≤ 3, χ′(G) = ∆, hence χ′′

a(G) ≤ χ(G) + χ′(G) ≤ ∆ + 3.

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Theorem 3.1.1 Let G be a 2-connected outerplanar graph. (1) If ∆ = 3, then χ′′

a(G) = 5 .

(2) If ∆ = 4, then 5 ≤ χ′′

a(G) ≤ 6; and χ′′ a(G) = 6

⇔ G has adjacent ∆-vertices.

[X.Chen, Z.Zhang, J. Lanzhou Univ. Nat. Sci. 42(2006) 96-102.]

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Theorem 3.1.2 If G is a 2-connected outerplanar graph with ∆ = 5, then 6 ≤ χ′′

a(G) ≤ 7; and

χ′′

a(G) = 7 ⇔ G has adjacent ∆-vertices.

[S.Zhang, X.Chen, X.Liu, Xibei Shifan Daxue Xuebao Ziran Kexue Ban, 41(5)(2005) 8-13.]

Theorem 3.1.3 If G is a 2-connected outerplanar graph with ∆ = 6, then 7 ≤ χ′′

a(G) ≤ 8; and

χ′′

a(G) = 8 ⇔ G has adjacent ∆-vertices.

[M.An, Hexi Xueyuan Xuebao 21(5)(2005) 25-29.]

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Theorem A Let G be an outerplane graph with ∆ ≥ 3. Then ∆+1 ≤ χ′′

a(G) ≤ ∆+2; and χ′′ a(G) =

∆ + 2 ⇔ G has adjacent ∆-vertices.

[Y.Wang, W.Wang, Adjacent vertex distinguishing total colorings of outerplanar graphs, J. Comb. Optim., 19(2010) 123-133.]

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A graph G has a graph H as a minor if H can be ob- tained from a subgraph of G by contracting edges, and G is called H-minor free if G does not have H as a minor. Theorem B Let G be a K4-minor free graph with ∆ ≥ 3. Then ∆+1 ≤ χ′′

a(G) ≤ ∆+2; and χ′′ a(G) =

∆ + 2 ⇔ G has adjacent ∆-vertices.

[W.Wang, P.Wang, Adjacent vertex distinguishing total colorings of K4- minor free graphs, Sci. China Ser.A., 39(2)(2009) 1462-1467.]

Since outerplanar graphs are K4-minor free graphs, Theorem B generalizes Theorem A.

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(3.2) χ′′

a for graphs with lower maximum aver-

age degree The maximum average degree mad(G) of a graph G is defined by mad(G) = max

H⊆G{2|E(H)|/|V (H)|}.

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Theorem C Let G be a graph with mad(G) = M. (1) If M < 8

3 and ∆ = 3, then χ′′ a(G) ≤ 5.

(2) If M < 3 and ∆ = 4, then χ′′

a(G) ≤ 6.

(3) If M < 3 and ∆ ≥ 5, then ∆ + 1 ≤ χ′′

a(G) ≤

∆ + 2; and χ′′

a(G) = ∆ + 2 ⇔ G has adjacent ∆-

vertices.

[W.Wang, Y.Wang, Adjacent vertex distinguishing total colorings of graphs with lower average degree, Taiwanese J. Math., 12(2008) 979- 990.]

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Girth g: the length of a shortest cycle in G Let G be a planar graph. Then mad(G) < 2g g − 2.

If G is planar and g ≥ 6, then mad(G) < 3;
If G is planar and g ≥ 8, then mad(G) < 8

3.

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Corollary C Let G be a planar graph. (1) If g ≥ 8 and ∆ = 3, then χ′′

a(G) ≤ 5.

(2) If g ≥ 6 and ∆ = 4, then χ′′

a(G) ≤ 6.

(3) If g ≥ 6 and ∆ ≥ 5, then ∆ + 1 ≤ χ′′

a(G) ≤

∆ + 2; and χ′′

a(G) = ∆ + 2 ⇔ G has adjacent ∆-

vertices.

[W.Wang, Y.Wang, Taiwanese J. Math., 12(2008) 979-990.]

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(3.3) χ′

a for graphs with lower maximum av-

erage degree (including planar graphs of high girth) Theorem D If G is a planar graph with g ≥ 6, then χ′

a(G) ≤ ∆ + 2.

[Y.Bu, K.Lih, W.Wang, Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least six, Discuss. Math. Graph Theory, to appear.]

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Theorem E Let G be a graph with mad(G) = M. (1) If M < 3 and ∆ ≥ 3, then χ′

a(G) ≤ ∆ + 2.

(2) If M < 7

3 and ∆ = 3, then χ′ a(G) ≤ 4.

(3) If M < 5

2 and ∆ ≥ 4, then χ′ a(G) ≤ ∆ + 1.

(4) If M < 5

2 and ∆ ≥ 5, then ∆ ≤ χ′ a(G) ≤ ∆+1;

and χ′

a(G) = ∆ + 1 ⇔ G has adjacent ∆-vertices.

[W.Wang, Y.Wang, Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree, J. Comb. Optim., 19(2010) 471-485.]

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Corollary E Let G be a planar graph. (1) If g ≥ 6 and ∆ ≥ 3, then χ′

a(G) ≤ ∆ + 2.

(2) If g ≥ 10 and ∆ ≥ 4, then χ′

a(G) ≤ ∆ + 1.

(3) If g ≥ 14 and ∆ = 3, then χ′

a(G) ≤ 4.

(4) If g ≥ 10 and ∆ ≥ 5, then ∆ ≤ χ′

a(G) ≤ ∆+1;

and χ′

a(G) = ∆ + 1 ⇔ G has adjacent ∆-vertices.

[W.Wang, Y.Wang, J. Comb. Optim., 19(2010) 471-485.]

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(3.4) χ′

a for K4-minor free graphs (including

uterplanar graphs)

Theorem F Let G be a K4-minor free graph. (1) If ∆ ≥ 4, then ∆ ≤ χ′

a(G) ≤ ∆ + 1.

(2) If ∆ ≥ 5, then χ′

a(G) = ∆+1 ⇔ G has adjacent

∆-vertices. Corollary F Let G be an outerplanar graph. (1) If ∆ ≥ 4, then ∆ ≤ χ′

a(G) ≤ ∆ + 1.

(2) If ∆ ≥ 5, then χ′

a(G) = ∆+1 ⇔ G has adjacent

∆-vertices.

[W.Wang, Y.Wang, Adjacent vertex distinguishing edge-colorings of K4- minor free graphs, submitted.]

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4

Outline of Proofs (4.1) Proof of Theorem A Lemma 1 Every outerplane graph G with |G| ≥ 2 contains one of (C1)-(C5) as follows:

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¶ (C1) a leaf is adjacent to a 3−-vertex. ¶ (C2) a path x1x2 · · · xn, with n ≥ 4, d(x1) = 2,

d(xn) = 2, d(x2) = · · · = d(xn−1) = 2.

¶ (C3) a 4+-vertex v is adjacent to a leaf and d(v)−

3 2−-vertices.

¶ (C4) a 3-face [uv1v2] with d(u) = 2, d(v1) = 3. ¶ (C5) two 3-faces [u1v1x] and [u2v2x] with d(x)

= 4, d(u1) = d(u2) = 2.

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✈ ✈ ✈ ✈

x2 x1 xn

❅

❅ ❅ ❅

(C2)

✈ ✈ ✈ ❆ ❆❆ ✁ ✁ ✁ ✁ ✁ ✁ ✈ ✈ ✈

u v1 v2 z y2 y1 (C4)

✈ ✈ ✈ ✈ ✈ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁

x v1 v2 u1 u2 (C5)

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Lemma 2 Every outerplane graph G with ∆ ≤ 3 contains one of (B1)-(B3):

¶ (B1) a vertex adjacent to at most one vertex that

is not a leaf.

¶ (B2) a path x1x2x3x4 such that each of x2 and x3

is either a 2-vertex, or a 3-vertex that is adjacent to a leaf.

¶ (B3) a 3-face [uxy] such that either d(u) = 2, or

d(u) = 3 and u is adjacent to a leaf.

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(4.2) Proof of Theorem B

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Let H be the graph obtained by removing all leaves

f G. Then mad(H) ≤ mad(G) < 3. H has the

following properties:

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Claim 7 (1) δ(H) ≥ 2; (2) If 2 ≤ dG(v) ≤ 3, then dH(v) = dG(v); (3) If dH(v) = 2, then dG(v) = 2; (4) If dG(v) ≥ 4, then dH(v) ≥ 3. We make use of discharging method. First, we de- fine an initial charge function w(v) = dH(v) for every v ∈ V (H).

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Next, we design a discharging rule and redis- tribute weights accordingly. Once the discharging is finished, a new charge function w′ is produced. However, the sum of all charges is kept fixed when the discharging is in progress. Nevertheless, we can show that w′(v) ≥ 3 for all v ∈ V (H). This leads to the following obvious contradiction: 3 = 3|V (H)|

|V (H)| ≤

v∈V (H) w′(v)

|V (H)|

=

v∈V (H) w(v)

|V (H)|

= 2|E(H)|

|V (H)| ≤ mad(H) < 3.

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