Adjacent Vertex Distinguishing Colorings of Graphs

Transcription

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

2 Our Page. 2 Total 80

3 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

4 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

5 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)}.

6 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

7 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

8 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

9 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

10 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.

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

12 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

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

14 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

15 (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

16 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) ] Page. 16 Total 80

17 Theorem For n 3, χ s(k n ) = n if n is odd, and χ s(k n ) = n + 1 if n is even. Theorem 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) ]

18 Theorem 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 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) ]

19 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 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) ] Page. 19 Total 80

20 Theorem 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 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 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) ]

21 Theorem 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 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) ]

22 Theorem Let G be disjoint union of sufficiently many k-regular 1-factorizable graphs. Then χ s(g) µ(g) + 1. Theorem 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 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) ] Page. 22 Total 80

23 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 For any integer k χ s(g), G has an equitable strong edge k-coloring. [J.Rudašová, R.Soták, Discrete Math., 308(2008) ] Page. 23 Total 80

24 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 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) ] Page. 24 Total 80

25 Theorem 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 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) ] Page. 25 Total 80

26 Theorem 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) ] Page. 26 Total 80

27 Theorem 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) ] Theorem 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) ] Page. 27 Total 80

28 (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 (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

29 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

30 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) ] Page. 30 Total 80

31 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) ] Page. 31 Total 80

32 Theorem For any graph G, χ a(g) 3. [S.Akbari, H.Bidkhori, N.Nosrati, Discrete Math. 306(2006) ] [M.Ghandehari, H.Hatami, Two upper bounds for the strong edge chromatic number, preprint.] Theorem For any graph G, χ a(g) 3 1. [Y.Dai, Y.Bu, Math. Econ., 26(1)(2009) ] Page. 32 Total 80

33 Theorem For a graph G with 10 6, we have χ a(g) + 27 ln. Theorem 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

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

35 Theorem If G is a graph with = 3, then χ a(g) 5. Theorem If G is a bipartite graph, then χ a(g) + 2. Theorem 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) ] Page. 35 Total 80

36 Theorem If G is a graph with = 4, then χ a(g) 8. [Y.Dai, Master Thesis, 2007] Theorem Let r 4. Then a random r- regular graph G asymptotically almost surely has χ a(g) 3r/2. Corollary 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

37 Theorem If G is a planar bipartite graph with 12, then χ a(g) + 1. Corollary 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) ] Question 2.4 Is necessary Corollary ? (If such graph has no adjacent -vertices, then χ a(g) =?) Page. 37 Total 80

38 Theorem If G is a Hamiltonian graph with χ(g) 3, then χ a(g) + 3. Theorem 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 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) ] Page. 38 Total 80

39 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 χ a(q p ) = p + 1 for all p 3. [M.Chen, X.Guo, Inform. Process. Lett., 109(2009) ] Page. 39 Total 80

40 Conjecture 2.5 (TCC) [M.Behzad 1965; V,G.Vizing, 1968] For a simple graph G, + 1 χ (G) + 2. Theorem For any simple graph G, χ (G) [M.Molloy, B.Reed, Combinatorics, 18(1998), ] Page. 40 Total 80

41 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) ] Page. 41 Total 80

42 Theorem 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) ] Page. 42 Total 80

43 (2.3) AVD Total Coloring Conjecture 2.8 For a graph G with n 2 vertices, χ a(g) 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) ]

44 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) ] Page. 44 Total 80

45 Theorem 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) = + 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

46 Theorem If G is a graph with = 3, then χ a(g) 6. [X.Chen, Discrete Math. 308(2008) ] [H.Wang, J. Comb. Optim. 14(2007) ] [J.Hulgan, Discrete Math. 309(2009) ] Question 2.9 [J.Hulgan, 2009] For a graph G with = 3, is the bound χ a(g) 6 sharp? Page. 46 Total 80

47 Theorem 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) ] 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 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) ]. Page. 47 Total 80

48 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

49 Theorem 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) ] Page. 49 Total 80

50 Theorem 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 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) ] Page. 50 Total 80

51 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) ] Page. 51 Total 80

52 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) ] Since outerplanar graphs are K 4 -minor free graphs, Theorem B generalizes Theorem A. Page. 52 Total 80

53 (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

54 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) ] Page. 54 Total 80

55 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

56 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) ]

57 (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

58 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) ] Page. 58 Total 80

59 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) ] Page. 59 Total 80

60 (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

61 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

62 (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

63 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

64 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

65 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

66 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

67 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

68 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

69 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

70 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

71 (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

72 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

73 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

74 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

75 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

76 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

77 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) = = 3.

78 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) = 3. Page. 78 Total 80

79 If d H (v) = 5, then v is adjacent to at most four 2-vertices by Claim 6. Thus, w (v) = 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

80 Thanks for Your Attention! Page. 80 Total 80