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b-coloring of Graphs S.Francis Raj, Post Doctoral Fellow, Deptartment of Mathematics, Bharathidasan University, Tiruchirappalli-620024. 1 Outline Preliminaries Bounds for the b-chromatic no. of G-v - Bounds for b(G-v) in terms of b(G) - Extremal graphs Bounds for the b-chromatic no. of the Myc. - Bounds for b(µ(G)) - Bounds for b(µm(Kn)) Scope for further research 2 b-coloring The b-chromatic number was introduced by R.W. Irving and D.F. Manlove. b-chromatic number has received wide attention from the time of its introduction. They have shown that the determination of b(G) is NP-hard for general graphs, but polynomial for trees. Here is a motivation as to why we study b-coloring. The motivation, is similar to achromatic number. 3 Motivation All graphs considered are simple, undirected and finite. A proper k-coloring is a map f : V S, where S is a set of distinct colors such that adjacent vertices receive distinct colors. Equivalently, A proper k-coloring is partitioning of the vertex set into independent sets v1 v2 v6 v7 v1 v2 v3 v5 v4 v7 v5 v8 v6 v9 v4 v3 v9 v8 4 Chromatic number Chromatic number: The chromatic number (G) of a graph G is the minimum number of colors needed for a proper vertex coloring of G. 5 Origin of b-coloring complete coloring: edge between every pair of color classes. A coloring c is said to be complete if no two classes can be combined together (without disturbing the proper coloring). Equivalently, Achromatic number = Max {|c|: c is complete}. 6 Origin of b-coloring A coloring c is said to be b-coloring if no classes can be distributed among the others (without disturbing the proper coloring). Equivalently, b-chromatic number = Max {|c|: c is b-coloring}. 7 Preliminaries b-chromatic number A proper k-coloring of a graph G is a b-coloring of G using k colors if each color class contains a color dominating vertex (c.d.v.), that is, a vertex adjacent to at least one vertex of every other color class. The b- chromatic number of a graph G, denoted by b(G), is the maximum k such that G has a b-coloring using k colors. There are at least b(G) vertices with degree at least b(G)-1. (G)≤ b(G) ≤ (G) +1 8 Part I - Bounds for b(G-v) and Extremal graphs 9 Bounds for b(G-v) Let v be any vertex of a graph G. (G-v)= (G) or (G)-1. Similarly for the achromatic number (G), (G-v)= (G) or (G)-1. Surprisingly, a similar statement does not hold good for the b-chromatic number b(G) of G. Indeed, the gap between b(G-v) and b(G) can be arbitrarily large. 10 Bounds for b(G-v) b(G-v) can arbitrarily be greater than b(G). u1 u2 u3 uk v1 v2 v3 vk G-v1 G b(G)=2 b(G-v1)=k 11 Bounds for b(G-v) b(G-v) can arbitrarily be lesser than b(G). u1 u2 u3 uk v v1 v2 v3 vk G -v1 G b(G)=k+1 b(G-v1)=2 12 Bounds for b(G-v) Aim: To find bounds for b(G-v) in terms of b(G). 13 Bounds for b(G-v) Theorem 3.1: Let G be a connected graph. Then b(G) = 2 iff G is bipartite and has a full vertex in each part (a vertex vX is said to be a full vertex in a bipartite graph G=(X,Y) if NG(v)=Y). Proof: X 2 Y 1 14 Bounds for b(G-v) x1 x2 X Y Y0 y1 y2 yk V1 V2 Vk 15 Bounds for b(G-v) Theorem 3.2: For any connected graph G with n≥5 vertices and for any vV(G), n n b(G ) 2 b(G v) b(G ) 2. 2 2 16 Extermal Graphs Graphs G for which b(G − v) attains the upper bound. Case (i) n is even n b(G v) b(G ) 2, n 6. 2 v is adjacent to all the classes of c (a b - chromatic coloring of G - v) S Singleton classes of c. T Other classes . Clearly | S | | T | b(G v). n | S | b(G ) 1 and thus | T | 1. 2 | V (T ) | n 2. 17 Extermal Graphs | V (T ) | n 2 | S | 0 and hence | G v | n, . n Thus | V (T ) | n 2, | T | 1 and thus | S | 1. 2 b(G ) 2 G-v S v 18 Extermal Graphs Case (ii) n is odd w u1 u2 u3 ub (G v ) 2 u1 u2 u3 ub (G v )1 x y v1 v2 v3 vb (G v ) 2 x v1 v2 v3 vb (G v )1 v Figure 3 v |V(T')|=n-3 |V(T')|=n-2 19 Extermal Graphs Case (ii) n is odd w u1 u2 u3 ub (G v )1 u1 u2 u3 u4 ub (G v )1 x v1 v2 v3 vb ( G v )1 v1 v2 v3 v4 vb ( G v )1 v v |V(T')|=n-2 |V(T')|=n-1 20 Extermal Graphs Case (ii) n is odd There are 4 families of graphs. For graphs G for which b(G − v) attains the lower bound, it is the other way. n is even : There are 4 families. n is odd : There is 1 family. 21 Part II - Bounds for b(µ(G)) 22 Mycielskian of a graph For a graph G=(V,E), the Mycielskian of G is the graph (G) with vertex set V V ' u, where V'= {x' : x V} and edge set E {xy' : xy E} {y'u : y' V'}. The vertex x' is called the twin of the vertex x (and x the twin of x') and the vertex u is called the root of (G). For n ≥ 2, n(G) is defined iteratively by setting n(G) = (n-1(G)) a a' a' a b b' u u c c' V V' Grötzsch graph 23 Observations d(G)(v)=2dG(v) for vV(G). d(G)(v)=dG(v)+1 for vV(G). d(G)(u)=|V(G)|. 24 Bounds for b(µ(G)) Aim: Our intension was to answer the following question: Let a, b, c, d be positive integers such that a<b<c<d. Are there graphs G with (G)=a, (G)=b, b(G)=c, and (G)+1=d? 25 Bounds for b(µ(G)) (G ) K c ,c {1F } Ka 26 Bounds for b(µ(G)) Mycielskians form an important family of graphs while considering graph coloring. Our first part of this chapter would be to find bounds for the b-chromatic number of the Mycielskian of some families of graphs. Theorem 2.1: Let G be a graph with b(G)=b, and let K(G) denote the set of vertices of degree at least b. If |K(G)| ≤ 2b-2, then b+1≤b(µ(G))≤ 2b-1. Let L(G)=V(G)-K(G). 27 Bounds for b(µ(G)) If x and x' are c.d.v. of distinct color classes, then x and u must belong to the same color class. x x' 2 K(G) K' d ≥2b d ≥b+1 1 L(G) L' d≤2b-2 d≤b 28 Bounds for b(µ(G)) Since by our assumption |K| ≤ 2b-2, there can be at most 2b-2+1=2b-1 c.d.v.'s of (G) in K K‘ {u} belonging to distinct color classes in (G). x L(G) L'(G), d(G)(x) ≤ (b-1)+(b-1)=2b-2. (L(G)= V(G)\K(G)) Thus b((G)) ≤ 2b(G)-1. 29 Bounds for b(µ(G)) As a consequence of Theorem 2.1, it follows that b(G)+1≤b(µ(G))≤ 2b(G)-1 for G in any of the following families: -- the hypercubes Qp, where p≥3. -- trees. -- a special class of bipartite graphs. -- regular graphs with girth at least 6. In fact, all graphs G with b(G)=(G)+1, fall into this category. -- split graphs. -- Kn,n-{a 1-factor}. 30 Bounds for b(µ(G)) All values in [b(G)+1,2b(G)-1] are attainable. v1 , 2 q 3 v2,2q 3 v1 , 2 v1 v1 , 2 q 2 w1 v2,2 v2 v2,2q 2 w2 v1,1 v 2 ,1 vq ,2 q 3 u 1, q 2 u p ,q 2 vq,2 vq vq ,2 q 2 wq u1, 2 u1 u 1 , q 1 w p q 1 u p ,2 u p u p , q 1 v q ,1 u 1,1 u p ,1 31 Bounds for b(µm(Kn)) Generalized Mycielskian 32 Bounds for b(µm(Kn)) Finally we look at the bounds for the generalized Mycielskian of Kn. We show that these bounds are sharp. The general upper bound for the b-chromatic number of µm(Kn), namely, (µm(Kn))+1=2n-1 also turns out to be sharp. Theorem 2.2: Let m be a positive integer such that 3≤m≤2n-1, where n≥2 is any integer. Then, Moreover, if m≥2n-1, b(µm(Kn))=2n-1. 33 Bounds for b(µ(G)) V0 V1 V2 V3 V4 V5 V6 V7 V8 V9 0 th row 1 1 4 4 18 18 10 10 24 24 2 2 5 5 19 19 11 11 25 25 3 3 6 6 20 20 12 12 12 12 x 3 rd row 4 15 15 1 1 15 15 1 1 15 5 16 16 2 2 16 16 2 2 16 6 17 17 3 3 17 17 3 3 17 7 18 18 7 7 4 4 7 7 4 1 8 19 19 8 8 5 5 8 8 5 9 20 20 9 9 6 6 9 9 6 10 21 21 10 10 21 21 18 18 21 11 22 22 11 11 22 22 19 19 22 12 23 23 12 12 23 23 20 20 23 13 24 24 13 13 24 24 13 13 10 ( n 1) th row 14 25 25 14 14 25 25 14 14 11 34 Bounds for b(µ(G)) V0 V1 V2 V3 V4 V5 V6 V7 V8 0 th row 1 1 5 5 22 22 13 13 13 2 2 6 6 23 23 14 14 22 3 3 7 7 24 24 15 15 15 4 4 8 8 25 25 16 16 16 x 4 th row 5 18 18 1 1 18 18 1 1 6 19 19 2 2 19 19 2 2 7 20 20 3 3 20 20 3 3 8 21 21 4 4 21 21 4 4 9 22 22 9 9 5 5 9 9 30 10 23 23 10 10 6 6 10 10 11 24 24 11 11 7 7 11 11 (qn 1 ) x th row 12 25 25 12 12 8 8 12 12 13 26 26 13 13 26 26 17 17 14 27 27 14 14 27 27 23 23 15 28 28 15 15 28 28 24 24 16 29 29 16 16 29 29 25 25 ( n 1) th row 17 30 30 17 17 30 30 30 26 35 Scope for further research 36 Does there exist graph G with b((G)) ≥ 2b(G)? Is |K(G)| ≤ 2b(G)-2 for chordal graphs G? For which graphs G, b(G)-1≤ b(G-v) ≤ b(G), for any vV(G)? Is it true that (G) is b-continuous whenever G is b- continuous? (G is called b-continuous if there exists a b-coloring of G using k colors for every k [(G),b(G)]) 37 Let a, b, c, d, and e be positive integers such that a<b<c<d<e. Are there graphs G with (G)=a, (G)=b, b(G)=c, (G)=d, and (G)+1=e? Characterize graphs G for which b(GH)=max{b(G),b(H)}, where denotes the cartesian product of G and H. (In general, b(GH) ≥ max {b(G),b(H)}). 38 References R. Balakrishanan and S. Francis Raj, Bounds for the b -chromatic number of the Mycielskian of some families of graphs, submitted. R. Balakrishanan and S. Francis Raj, Bounds for the b -chromatic number of the Mycielskian of some families of graphs: II, submitted. R. Balakrishanan and S. Francis Raj, Bounds for the b -chromatic number of vertex-deleted subgraphs and extremal graphs, Electronic Notes in Discrete Mathematics 34 (2009) 353–358. 39 References R. Balakrishanan and S. Francis Raj, R. Balakrishanan and S. Francis Raj, Bounds for the b -chromatic number of G-v, submitted. D. Barth, J. Cohen, T. Faik, On the b-continuity property of graphs, Discrete Appl. Math. 155 (2007) 1761-1768. F. Bonomo, G. Duran, F. Maffray, J. Marenco, and M.V. Pabon, On the b-coloring of cographs and P4-sparse graphs., graphs and combinatorics. S. Corteel, M. Valencia-Pabon and J.C. Vera, On approximating the b-chromatic number, Discrete Appl. Math. 146 (2005) 106-110. 40 References B. Effantin, H. Kheddouci, The b-chromatic number of some power graphs, Discrete Math. Theor. Comput. Sci. 6 (2003) 45-54. T. Faik, About the b-continuity of graph, Electronic Notes in Discrete Mathematics, 17 (2004) 151-156. H. Hajibolhassan, On the b-chromatic number of Kneser graphs, Discrete Applied Mathematics, (2009). C.T.Hoang, and M. Kouider, On the b-dominating coloring of graphs, Discrete Applied Mathematics, 152 (2005) 176-186. 41 References R.W. Irving and D.F. Manlove, The b-Chromatic number of a graph, Discrete Applied Mathematics, 91 (1999) 127-141. S. Klavzar and M. Jakovac, The b-chromatic number of cubic graphs, personal communication. M. Kouider and M. Maheo, Some bounds for the b- Chromatic number of a graph, Discrete Math. 256 (2002) 267-277. M. Kouider and M. Maheo, The b-chromatic number of the Cartesian product of two graphs, Studia Scientiarum Mathematicarum Hungarica 44 (2007) 49-55. 42 References M. Kouider, M. Zaker, Bounds for the b-Chromatic number of some families of graphs, Discrete Math. 306 (2006) 617--623. J. Kratochvil, Z. Tuza, and M. Voigt, On the b- chromatic number of graphs, Lecture Notes in Comput. Sci. 2573 (2002) 310-320. F. Maffray and M. Mechebbek, On b-perfect graphs, graphs and combinatorics 25 (2009) 365-375. B. Omoomi and R. Javadi, On the b-coloring of Cartesian product of graphs, to appear. B. Omoomi and R. Javadi, On b-coloring of the Kneser graphs, Discrete Mathematics (2009). 43 References P.C.B. Lam, G. Gu, W. Lin, Z. Song, Some properties of generalized Mycielski’s graphs, manuscript. P.C.B. Lam,G. Gu, W. Lin, Z. Song, Circular chromatic number and a generalization of the construction of Mycielski, J. Combin. Theory Ser. B, 89, (2003) 195–205. 44 45 Preliminaries Chromatic number: The chromatic number (G) of a graph G is the minimum number of colors needed for a proper vertex coloring of G. -coloring (-1)-coloring, a contradiction. For any chromatic coloring between any two classes there is an edge, that is, The minimum with this property --- chromatic number The maximum with this property --- achromatic number 46 Preliminaries Chromatic number: The chromatic number (G) of a graph G is the minimum number of colors needed for a proper vertex coloring of G. -coloring (-1)-coloring, a contradiction. For any chromatic coloring every color class contains a color dominating vertex (c.d.v.). The minimum with this property --- chromatic number 47 Results (G)≤ (G). There are at least b(G) vertices with degree at least b(G)-1. (G)≤ b(G) ≤ (G) +1. (G) ≤ (G) +1 need not be true. 48