bcoloring of Graphs

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b-coloring of Graphs

• S.Francis Raj,
• Post Doctoral Fellow,
• Deptartment of Mathematics,
• Bharathidasan University,
• Tiruchirappalli-620024.

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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

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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.

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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 v5 v8
v2 v4 v6
v3 v7 v9
v5
v4
v9
v3
v8
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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.

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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.
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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.
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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
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Part I - Bounds for b(G-v) and Extremal graphs
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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.

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Bounds for b(G-v)

• b(G-v) can arbitrarily be greater than b(G).

G b(G)2
G-v1 b(G-v1)k
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Bounds for b(G-v)

• b(G-v) can arbitrarily be lesser than b(G).

G b(G)k1
G-v1 b(G-v1)2
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Bounds for b(G-v)

• Aim
• To find bounds for b(G-v) in terms of b(G).

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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 v?X is said to be a full vertex in a
  bipartite graph G(X,Y) if NG(v)Y).
• Proof

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X
Y
1
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Bounds for b(G-v)
x1
x2
X
Y
Y0
y1
y2
yk
V1
V2
Vk
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Bounds for b(G-v)

• Theorem 3.2
• For any connected graph G with n5 vertices
  and for any v?V(G),

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Extermal Graphs

• Graphs G for which b(G - v) attains the upper
  bound.
• Case (i) n is even

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Extermal Graphs
G-v
S
v
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Extermal Graphs
Case (ii) n is odd
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Extermal Graphs
Case (ii) n is odd
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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.

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Part II - Bounds for b(µ(G))
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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'
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Grötzsch graph
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Observations

• d?(G)(v)2dG(v) for v?V(G).
• d?(G)(v?)dG(v)1 for v?V(G).
• d?(G)(u)V(G).

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Bounds for b(µ(G))

• Aim
• Our intension was to answer the following
  question
• Let a, b, c, d be positive integers such that
  altbltcltd. Are there graphs G with ?(G)a, ?(G)b,
  b(G)c, and ?(G)1d?

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Bounds for b(µ(G))
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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 b1b(µ(G)) 2b-1.
• Let L(G)V(G)-K(G).

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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' d b1
K(G) d 2b
1
L(G) d2b-2
L' d b
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Bounds for b(µ(G))

• Since by our assumption K 2b-2, there can be
  at most 2b-212b-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.

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Bounds for b(µ(G))

• As a consequence of Theorem 2.1, it follows that
  b(G)1b(µ(G)) 2b(G)-1 for G in any of the
  following families
• -- the hypercubes Qp, where p3.
• -- 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.

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Bounds for b(µ(G))

• All values in b(G)1,2b(G)-1 are attainable.

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Bounds for b(µm(Kn))

• Generalized Mycielskian

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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))12n-1 also turns out to be sharp.
• Theorem 2.2
• Let m be a positive integer such that 3m2n-1,
  where n2 is any integer. Then,
• Moreover, if m2n-1, b(µm(Kn))2n-1.

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Bounds for b(µ(G))
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Bounds for b(µ(G))
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Scope for further research
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• 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 v?V(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))

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• Let a, b, c, d, and e be positive integers such
  that altbltcltdlte. Are there graphs G with ?(G)a,
  ?(G)b, b(G)c, ?(G)d, and ?(G)1e?
• Characterize graphs G for which
  b(G?H)maxb(G),b(H), where ? denotes the
  cartesian product of G and H. (In general,
  b(G?H) max b(G),b(H)).

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• 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) 353358.

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• 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.

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• 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.

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• 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.

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• 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).

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• References
• P.C.B. Lam, G. Gu, W. Lin, Z. Song, Some
  properties of generalized Mycielskis 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) 195205.

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Thank you
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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
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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
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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.