S.Francis Raj-2

					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
  vX 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 vV(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 vV(G).
d(G)(v)=dG(v)+1 for vV(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
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)])
                                                 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(GH)=max{b(G),b(H)}, where  denotes
the cartesian product of G and H. (In
general, b(GH) ≥ 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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:36
posted:9/29/2012
language:Unknown
pages:48