Adjacent Vertex Distinguishing Colorings of Graphs

Adjacent Vertex Distinguishing Colorings of Graphs Wang Weifan Department of Mathematics Zhejiang Normal University Jinhua 321004 Page. 1 Total 80

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1 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}. Page. 3 Total 80

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}. Page. 4 Total 80

For an edge coloring f of G and for a vertex v V, we define: C f (v) = {f(e) e is incident to v}. For a total coloring f of G and for a vertex v V, we define: Page. 5 Total 80 C f [v] = {f(e) e is incident to v} {f(v)}.

Vertex-Distinguishing edge coloring (VD edge coloring) or Strong edge coloring: A proper edge coloring f such that C f (u) C f (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}. Page. 6 Total 80

Vertex-Distinguishing total coloring (VD total coloring) or Strong total coloring:: A proper total coloring f such that C f [u] C f [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}. Page. 7 Total 80

Adjacent-Vertex-Distinguishing edge coloring (AVD edge coloring): A proper edge coloring f such that C f (u) C f (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}. Page. 8 Total 80

Adjacent-Vertex-Distinguishing total coloring (AVD total coloring): A proper total coloring f such that C f [u] C f [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}. Page. 9 Total 80

Examples First Example: C 5 χ (C 5 ) = 3, χ a(c 5 ) = 5, Page. 10 Total 80 χ a(c 5 ) = χ (C 5 ) = 4.

1 2 1 3 2 4 4 5 3 1 1 4 2 4 3 χ ' ( C a 5) = 5 χ " ( C ) = 4 a 5 Page. 11 Total 80

Second Example: K 4 χ (K 4 ) = 3, χ a(k 4 ) = 5, χ a(k 4 ) = χ (K 4 ) = 5. Page. 12 Total 80

4 3 1 1 5 2 χ ' ( K ) = 5 a 4 1 5 2 4 2 3 1 4 3 5 χ ( K ) = 5 " a 4 Page. 13 Total 80

2 : 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 n d : the number of d-vertices in G Page. 14 Total 80

(2.1) Strong Edge Coloring (VD Edge Coloring) The concept of strong edge coloring was introduced independently by Aigner, Triesch, and Tuza, by Hornák and Soták, 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 min{k ( k) δ d d nd }. Page. 15 Total 80

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.] Page. 16 Total 80

Theorem 2.1.1 For n 3, χ s(k n ) = n if n is odd, and χ s(k n ) = n + 1 if n is even. Theorem 2.1.2 Let n be a cycle of length n 3 and let µ = µ(c n ). Then χ s(c n ) = µ + 1 if µ is odd and ( µ 2 ) 2 n ( µ2 ) 1 or µ is even and n > (µ 2 2µ)/2, and χ s(c n ) = µ otherwise. Page. 17 Total 80 [A.C.Burris and R.H.Schelp, J.Graph Theory, 26(1997) 73-82.]

Theorem 2.1.3 For a graph G, m 1 χ s(g) ( + 1) 2m 2 + 5, where m 1 = max {(k!n k) k 1 + k 1 1 k 2 }, m 2 = max 1 k n1 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. Page. 18 Total 80 [A.C.Burris,R.H.Schelp, J.Graph Theory, 26(1997) 73-82.]

Let c be the smallest number in the interval (4, 6.35) such that 6c 2 +c(49c 2 208) 1 2 c 2 16 < max{n 1 + 1, cn 1 2 2, 21}. Theorem 2.1.5 For a tree T K 2, we have χ s(t ) max{n 1 + 1, cn 1 2 2, 21}. [A.C.Burris, R.H.Schelp, J.Graph Theory, 26(1997) 73-82.] Page. 19 Total 80

Theorem 2.1.6 Let G be a vertex-disjoint union of cycles, and let k be the least number such that n ( k2 ). 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 n 1 k and n 2 ( 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 n 1 k and n 2 ( k 2 ). Then k χ s (G) k + 5. Page. 20 Total 80 [P.N.Balister, B.Bollbás, R.H.Schelp, Discrete Math., 252(2002) 17-29.]

Theorem 2.1.9 If L m is an m-sided prism, then χ s(l m ) µ(l m ) + 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(rk 4 ) µ(rk 4 ) + 1. Page. 21 Total 80 [K.Taczuk, M.Woźniak, Opuscula Math., 24/2(2004) 223-229.]

Theorem 2.1.11 Let G be disjoint union of sufficiently 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 positive integer r, χ s(g) µ(rg) + 1. Theorem 2.1.13 Let G {K 4,4, K 5,5, K 6,6, K 7,7, K 6 }. Then, for each integer r, χ s(g) µ(rg) + 1. [J.Rudašová, R.Soták, Discrete Math., 308(2008) 795-802.] Page. 22 Total 80

A proper edge k-coloring of a graph G is called equitable if the number of 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.Rudašová, R.Soták, Discrete Math., 308(2008) 795-802.] Page. 23 Total 80

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 v 1 and v 2 with the neighborhood of v in G equal to the disjoint union of the neighborhoods of v 1 and v 2 in G. Call a split G an r-split if the degree of v 1 (or v 2 ) 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.] Page. 24 Total 80

Theorem 2.1.16 If n = 1, n 2 = n 1 = 0, n 0, n 1, n 1 1, n 3, n 4 2, n 3 1, and for 5 d 4, n d d 4 d 3 min{2 then χ s(g) =. ( ) ( 5 2, 3 d ) } 2, Theorem 2.1.17 If G n,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(g n,p ) = ) 1 as n. [P.B.Balister, Random Struct. Alg., 20(2001) 89-97.] Page. 25 Total 80

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.] Page. 26 Total 80

Theorem 2.1.19 Let G be a graph on n 3 vertices. If δ > n 3, then χ s(g) + 5. [C.Bazgan, H.Li, M.Woźniak, 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.] Page. 27 Total 80

(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. Page. 28 Total 80

Conjecture 2.3 (Z.Zhang, L.Liu, J.Wang, 2002) For a normal graph G ( C 5 ), χ a(g) + 2. Note: χ a(c 5 ) = 5 = + 3. Is unique C 5 as an exception? Page. 29 Total 80

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 distinct degree, then χ a(g) =. For a cycle C n, χ a(c n ) = 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] Page. 30 Total 80

For a complete graph K n with n 3, χ a(k n ) = n if n 0 (mod 2), and χ a(k n ) = n + 1 if n 1 (mod 2). For a complete bipartite graph K m,n with 1 m n, χ a(k m,n ) = n if m < n, and χ a(k m,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] Page. 31 Total 80

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.] Page. 32 Total 80

Theorem 2.2.4 For a graph G with 10 6, we have χ a(g) + 27 ln. Theorem 2.2.5 For an -regular graph G, with > 100, we have χ a(g) + 3 log 2. [M.Ghandehari, H.Hatami, Two upper bounds for the strong edge chromatic number, preprint.] Page. 33 Total 80

Theorem 2.2.6 For a graph G with > 10 20, then χ a(g) + 300. [H.Hatami, J. Combin. Theory Ser. B, 95(2005) 246-256. ] Page. 34 Total 80

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.Győri, J.Lehel, R.H.Schelp, SIAM J. Discrete Math. 21(2007) 237-250.] Page. 35 Total 80

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.Rucński, The Electronic J. Combin., 13(2006), R77.] Page. 36 Total 80

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.Horňák, M.Woźniak, 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) =?) Page. 37 Total 80

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 dominating 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.] Page. 38 Total 80

The p-dimensional hypercube Q p is the graph whose vertices are the ordered p-tuples of 0 s and 1 s, two vertices being adjacent if and only if they differ in exactly one coordinate. For example, Q 2 is a 4-cycle, and Q 3 is the cube. Theorem 2.2.18 χ a(q p ) = p + 1 for all p 3. [M.Chen, X.Guo, Inform. Process. Lett., 109(2009) 599-602.] Page. 39 Total 80

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) + 10 26. [M.Molloy, B.Reed, Combinatorics, 18(1998), 214-280.] Page. 40 Total 80

Conjecture 2.6 For a graph G with no K 2 or C 5 component, χ a(g) χ (G). Conjecture 2.7 For a r-regular graph G with no C 5 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.] Page. 41 Total 80

Theorem 2.2.20 Conjecture 2.7 holds for all regular 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 C n 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.] Page. 42 Total 80

(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. Page. 43 Total 80 [Z.Zhang, X.Chen, J.Li, B.Yao, X.Lu, J.Wang, Sci. China Ser. A 34 (2004) 574-583]

If n 4, then χ a(c n ) = 4. a(k n ) = n + 1 if n is even, χ a(k n ) = n + 2 otherwise. χ Let n + m 2. Then χ a(k m,n ) = + 1 if m n, χ a(k m,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] Page. 44 Total 80

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. Page. 45 Total 80

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? Page. 46 Total 80

Theorem 2.3.3 Let Q p be a p-dimensional hypercube with p 2, then χ a(q p ) = 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]. Page. 47 Total 80

3 Our (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. Page. 48 Total 80

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.] Page. 49 Total 80

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.] Page. 50 Total 80

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.] Page. 51 Total 80

A graph G has a graph H as a minor if H can be obtained 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 K 4 -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 K 4 - minor free graphs, Sci. China Ser.A., 39(2)(2009) 1462-1467.] Since outerplanar graphs are K 4 -minor free graphs, Theorem B generalizes Theorem A. Page. 52 Total 80

(3.2) χ a for graphs with lower maximum average degree The maximum average degree mad(g) of a graph G is defined by mad(g) = max {2 E(H) / V (H) }. H G Page. 53 Total 80

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.] Page. 54 Total 80

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. Page. 55 Total 80

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. Page. 56 Total 80 [W.Wang, Y.Wang, Taiwanese J. Math., 12(2008) 979-990.]

(3.3) χ a for graphs with lower maximum average 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.] Page. 57 Total 80

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.] Page. 58 Total 80

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.] Page. 59 Total 80

(3.4) χ a for K 4 -minor free graphs (including outerplanar graphs) Theorem F Let G be a K 4 -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 K 4 - minor free graphs, submitted.] Page. 60 Total 80

4 (4.1) Proof of Theorem A Lemma 1 Every outerplane graph G with G 2 contains one of (C1)-(C5) as follows: Page. 61 Total 80

(C1) a leaf is adjacent to a 3 -vertex. (C2) a path x 1 x 2 x n, with n 4, d(x 1 ) 2, d(x n ) 2, d(x 2 ) = = d(x n 1 ) = 2. (C3) a 4 + -vertex v is adjacent to a leaf and d(v) 3 2 -vertices. (C4) a 3-face [uv 1 v 2 ] with d(u) = 2, d(v 1 ) = 3. (C5) two 3-faces [u 1 v 1 x] and [u 2 v 2 x] with d(x) = 4, d(u 1 ) = d(u 2 ) = 2. Page. 62 Total 80

u u 1 u 2 x 1 x 2 x n (C2) v 1 v 2 y 1 z y 2 (C4) v 1 v 2 x (C5) Page. 63 Total 80

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 x 1 x 2 x 3 x 4 such that each of x 2 and x 3 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. Page. 64 Total 80

Lemma 3 Every outerplane graph G with = 4 contains one of (A1)-(A4): (A1) a vertex v with d(v) 3 adjacent to a leaf. (A2) a 3-vertex adjacent to at least two leaves. (A3) a path x 1 x 2 x 3 x 4 such that each of x 2 and x 3 is either a 2-vertex, or a 3-vertex that is adjacent to a leaf. (A4) a 3-face [uxy] with d(x) = 3 such that either d(u) = 2, or d(u) = 3 and u is adjacent to a leaf. Page. 65 Total 80

Lemma 4 Every outerplane graph G with = 3 and without adjacent 3-vertices contains (D1) or (D2): (D1) a leaf. (D2) a cycle x 1 x 2 x n x 1, with n 3, d(x 1 ) = 3, d(x 2 ) = = d(x n 1 ) = 2. Page. 66 Total 80

Theorem A a(g) 5. χ If G is an outerplane graph with 3, then Proof: The proof proceeds by induction on σ(g) (= G + G ). If σ(g) 5, the theorem holds trivially. Suppose that G is an outerplane graph with 3 and σ(g) 6. By the induction assumption, any outerplane graph H with (H) 3 and σ(h) < σ(g) has a total-5-avd-coloring f. By Lemma 2, G contains one of (B1)-(B3). We reduce each possible case to extend f to the whole graph G. Page. 67 Total 80

Theorem A If G is an outerplane graph with = 3 and without adjacent 3-vertices, then χ a(g) = 4. Proof: By induction on σ(g). By Lemma 4, we handle possible case (D1) or (D2). Page. 68 Total 80

Theorem A If G is an outerplane graph with 4, then χ a(g) + 2. Proof: By induction on σ(g). By Lemma 1, we handle each possible case of (C1)-(C5). Page. 69 Total 80

Theorem A If G is an outerplane graph with 4 and without adjacent -vertices, then χ a(g) = + 1. Proof: By induction on σ(g). By Lemma 3, we handle each possible case of (A1)-(A4). Page. 70 Total 80

(4.2) Proof of Theorem B Theorem B If G is a graph with mad(g) < 3 and K(G) = max{ + 2, 6}, then χ a(g) K(G). Proof: The proof proceeds by induction on σ(g) (= G + G ). If σ(g) 5, the theorem holds trivially. Suppose that G is a graph with mad(g) < 3 and σ(g) 6. the induction assumption, any proper subgraph H of G has a total-k-avd-coloring f. By Page. 71 Total 80

Claim 1 No 3 -vertex is adjacent to a leaf. Claim 2 No path x 1 x 2 x n with d(x 1 ), d(x n ) 3, d(x 2 ) = = d(x n 1 ) = 2, where n 4. Claim 3 No k-vertex v, k 4, with neighbors v 1, v 2,, v k such that d(v 1 ) = 1, d(v i ) 2 for 2 i k 2. Page. 72 Total 80

Claim 4 No 2-vertex adjacent to a 3-vertex. Claim 5 No 4-vertex adjacent to three 2-vertices. Claim 6 No 5-vertex v adjacent to five 2-vertices. Page. 73 Total 80

Let H be the graph obtained by removing all leaves of G. Then mad(h) mad(g) < 3. H has the following properties: Page. 74 Total 80

Claim 7 (1) δ(h) 2; (2) If 2 d G (v) 3, then d H (v) = d G (v); (3) If d H (v) = 2, then d G (v) = 2; (4) If d G (v) 4, then d H (v) 3. We make use of discharging method. First, we define an initial charge function w(v) = d H (v) for every v V (H). Page. 75 Total 80

Next, we design a discharging rule and redistribute 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 V (H) V (H) = 2 E(H) V (H) 3 = v V (H) w (v) V (H) mad(h) < 3. = v V (H) w(v) V (H) Page. 76 Total 80

The discharging rule is defined as follows: (R) Every 4 + -vertex gives 2 1 vertex. to each adjacent 2- Let v V (H). So d H (v) 2 by Claim 7(1). If d H (v) = 2, then v is adjacent to two 4 + -vertices by Claim 4. By (R), Page. 77 Total 80 w (v) d H (v) + 2 1 2 = 2 + 1 = 3.

If d H (v) = 3, then w (v) = w(v) = 3. If d H (v) = 4, then v is adjacent to at most two 2- vertices by Claim 5. Thus, w (v) 4 2 1 2 = 3. Page. 78 Total 80

If d H (v) = 5, then v is adjacent to at most four 2-vertices by Claim 6. Thus, w (v) 5 4 1 2 = 3. If d H (v) 6, then v is adjacent to at most d H (v) 2-vertices and hence w (v) d H (v) 1 2 d H(v) = 1 2 d H(v) 3. Page. 79 Total 80

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