Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 279246, 5 pages
doi:10.1155/2011/279246
Research Article
Equitable Coloring on Total Graph of Bigraphs and
Central Graph of Cycles and Paths
J. Vernold Vivin,1 K. Kaliraj,2 and M. M. Akbar Ali3
1
Department of Mathematics, University College of Engineering Nagercoil, Anna University of Technology
Tirunelveli (Nagercoil Campus), Nagercoil 629 004, Tamil Nadu, India
2
Department of Mathematics, R.V.S College of Engineering and Technology, Coimbatore 641 402,
Tamil Nadu, India
3
Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore 641 062,
Tamil Nadu, India
Correspondence should be addressed to J. Vernold Vivin, vernoldvivin@yahoo.in
Received 2 December 2010; Accepted 9 February 2011
Academic Editor: Marco Squassina
Copyright q 2011 J. Vernold Vivin et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The notion of equitable coloring was introduced by Meyer in 1973. In this paper we
obtain interesting results regarding the equitable chromatic number χ for the total graph of
complete bigraphs T Km,n , the central graph of cycles CCn and the central graph of paths
CPn .
1. Introduction
The central graph 1, 2 CG of a graph G is formed by adding an extra vertex on each
edge of G, and then joining each pair of vertices of the original graph which were previously
nonadjacent.
The total graph 3, 4 of G has vertex set V G ∪ EG and edges joining all elements
of this vertex set which are adjacent or incident in G.
If the set of vertices of a graph G can be partitioned into k classes V1 , V2 , . . . , Vk such
that each Vi is an independent set and the condition ||Vi | − |Vj || ≤ 1 holds for every pair
i, j, then G is said to be equitably k-colorable. The smallest integer k for which G is equitable
k-colorable is known as the equitable chromatic number 5–10 of G and denoted by χ G.
Additional graph theory terminology used in this paper can be found in 3, 4.
2
International Journal of Mathematics and Mathematical Sciences
2. Equitable Coloring on Total Graph of Complete Bigraphs
Theorem 2.1. If m ≤ n, the equitable chromatic number of total graph of complete bigraphs Km,n ,
χ T Km,n ⎧
⎨n 1
if m < n,
⎩n 2
if m n.
2.1
Proof. Let X, Y be the bipartition of Km,n , where X {vi : 1 ≤ i ≤ m} and Y {vj : 1 ≤ j ≤ n}.
Let uij 1 ≤ i ≤ m; 1 ≤ j ≤ n be the edges of vi vj . By the definition of total graph, T Km,n has
the vertex set {vi : 1 ≤ i ≤ m} ∪ {vj : 1 ≤ j ≤ n} ∪ {uij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} and the vertices
{uij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} induce n disjoint cliques of order n in T Km,n . Also vi 1 ≤ i ≤ m
is adjacent to vj 1 ≤ j ≤ n.
Case 1 if m n, χ T Km,n n 2. Now we partition the vertex set V T Km,n as follows:
V1 u11 , u2n , u3n−1 , u4n−2 , . . . , un−13 , un2 ,
V2 u12 , u21 , u3n , u4n−1 , . . . , un−14 , un3 ,
..
.
Vn u1n , u2n−1 , u3n−2 , u4n−3 , . . . , un−13 , un1 ,
2.2
Vn1 {v1 , v2 , . . . , vn },
Vn2 v1 , v2 , . . . , vn .
Clearly V1 , V2 , . . . , Vn2 are independent sets and |Vi | n 1 ≤ i ≤ n2 satisfying the condition
||Vi | − |Vj || 0, for any i / j, χ T Km,n ≤ n 2. Since there exists a clique of order n 1 in
T Km,n . χT Km,n ≥ n 1, also each vi of T Km,n receives one color different from the color
class assigned to the clique induced by {uij : 1 ≤ i ≤ m; 1 ≤ j ≤ n}. By the definition of total
graph, each vi is adjacent with vj 1 ≤ j ≤ n. Therefore, {v1 , v2 , . . . , vm } and {v1 , v2 , . . . , vn }
are independent sets and hence χT Km,n ≥ n 2. That is, χ T Km,n ≥ χT Km,n ≥ n 2;
therefore χ T Km,n ≥ n 2. Hence χ T Km,n n 2.
Case 2 if m < n, χ T Km,n n 1. Now we partition the vertex set V T Km,n as follows:
V1 {u11 , u22 , u33 , u44 , . . . , umm } ∪ vn ,
V2 u12 , u23 , u34 , . . . , umm−1 ∪ {um1 } ∪ v1 ,
V3 u13 , u24 , u35 , . . . , umm−2 ∪ um−13 , um2 ∪ v2 ,
..
.
,
Vn−1 u1n−1 , u2n ∪ u31 , u32 , . . . , umm−2 ∪ vn−2
Vn {u1n } ∪ u21 , u32 , . . . , umm−1 ∪ vn−1 ,
Vn1 {v1 , v2 , v3 , . . . , vm }.
2.3
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3
Clearly V1 , V2 , . . . , Vn1 are independent sets of T Km,n . Also |V1 | |V2 | · · · |Vn | m 1
j, χ T Km,n ≤ n 1. Since
and |Vn1 | m satisfy the condition ||Vi | − |Vj || ≤ 1, for any i /
there exists a clique of order n 1 in T Km,n . χT Km,n ≥ n 1, that is, χ T Km,n ≥
χT Km,n ≥ n 1, therefore χ T Km,n ≥ n 1. Hence χ T Km,n n 1.
3. Equitable Coloring on Central Graph of Cycles and Paths
Theorem 3.1. If n ≥ 5, the equitable chromatic number of central graph of cycles Cn ,
⎧
n1
⎪
⎪
⎨
2
χ CCn ⎪
n
⎪
⎩
2
if n is odd,
3.1
if n is even.
Proof. Let V Cn {v1 , v2 , . . . , vn } and ECn {e1 , e2 , . . . , en } be the vertices and edges of Cn
taken in the cyclic order. By the definition of central graph, CCn has the vertex set V Cn ∪
{ui : 1 ≤ i ≤ n}, where ui is the vertex of subdivision of the edge ei and joining all the
nonadjacent vertices of Cn in CCn .
Case 1 n is odd. We partition the vertex set V CCn as
V1 {v1 , v2 , un−2 , un−1 },
V2 {v3 , v4 , un },
V3 {v5 , v6 , u1 , u2 },
V4 {v7 , v8 , u3 , u4 },
..
.
3.2
Vn−1/2 {vn−2 , vn−1 , un−6 , un−5 },
Vn1/2 {vn , un−4 , un−3 }.
Clearly V1 , V2 , . . . Vn−1/2 , Vn1/2 are independent sets of CCn . Also |V1 | |V3 | |V4 | · · · |Vn−1/2 | 4 and |V2 | |Vn1/2 | 3. The inequality ||Vi | − |Vj || ≤ 1 holds, for any
i/
j, χ CCn ≤ n 1/2. For each i, vi is nonadjacent with vi−1 and vi1 and hence
χCCn ≥ n 1/2. That is, χ CCn ≥ χCCn ≥ n 1/2, χ CCn ≥ n 1/2.
Therefore, χ CCn n 1/2.
Case 2 n is even. Now we partition the vertex set V CCn as follows:
V1 {v1 , v2 , un−3 , un−2 },
V2 {v3 , v4 , un−1 , un },
V3 {v5 , v6 , u1 , u2 },
V4 {v7 , v8 , u3 , u4 },
..
.
Vn/2 {vn−1 , vn , un−5 , un−4 }.
3.3
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International Journal of Mathematics and Mathematical Sciences
Clearly V1 , V2 , . . . Vn/2 are independent sets of CCn . Also |V1 | |V2 | |V3 | |V4 | · · · j, χ CCn ≤ n/2. For each i, vi
|Vn/2 | 4. The inequality ||Vi | − |Vj || 0 holds, for any i /
is nonadjacent with vi−1 and vi1 and hence χCCn ≥ n/2. That is, χ CCn ≥ χCCn ≥
n/2, χ CCn ≥ n/2. Therefore, χ CCn n/2.
Remark 3.2. If n 3, 4, then χ CCn 2, 3, respectively.
Theorem 3.3. If n ≥ 5, the equitable chromatic number of central graph of paths Pn ,
⎧n 1
⎪
⎨
2
χ CPn ⎪
n
⎩
2
if n is odd,
if n is even.
3.4
Proof. Let V Pn {v1 , v2 , . . . , vn } and EPn {e1 , e2 , . . . , en } be the vertices and edges of Pn .
By the definition of central graph, CPn has the vertex set V Pn ∪{ui : 1 ≤ i ≤ n−1}, where ui
is the vertex of subdivision of the edge ei and joining all nonadjacent vertices of Pn in CPn .
Case 1 n is odd. Now we partition the vertex set V CPn as follows:
V1 {v1 , v2 , un−2 },
V2 {v3 , v4 , un−1 },
V3 {v5 , v6 , u1 , u2 },
V4 {v7 , v8 , u3 , u4 },
..
.
3.5
Vn−1/2 {vn−1 , vn−2 , un−6 , un−5 },
Vn1/2 {vn , un−4 , un−3 }.
Clearly V1 , V2 , . . . Vn−1/2 , Vn1/2 are independent sets of CPn . Also |V3 | |V4 | · · · |Vn−1/2 | 4 and |V1 | |V2 | |Vn1/2 | 3. The inequality ||Vi | − |Vj || ≤ 1 holds, for
any i / j, χ CPn ≤ n 1/2. For each i, vi is nonadjacent with vi−1 and vi1 and hence
χCPn ≥ n 1/2. That is, χ CPn ≥ χCPn ≥ n 1/2, χ CPn ≥ n 1/2.
Therefore χ CPn n 1/2.
Case 2 n is even. Now we partition the vertex set V CPn as follows:
V1 {v1 , v2 , un−3 , un−2 },
V2 {v3 , v4 , un−1 },
V3 {v5 , v6 , u1 , u2 },
V4 {v7 , v8 , u3 , u4 },
..
.
Vn/2 {vn−1 , vn , un−5 , un−4 }.
3.6
International Journal of Mathematics and Mathematical Sciences
5
Clearly V1 , V2 , . . . Vn/2 are independent sets of CPn . Also |V1 | |V3 | |V4 | · · · |Vn/2 | 4
j, χ CPn ≤ n/2. For each i, vi is
and |V2 | 3. The inequality Vi | − |Vj ≤ 1 holds for any i /
nonadjacent with vi−1 and vi1 and hence χCPn ≥ n/2. That is, χ CPn ≥ χCPn ≥ n/2,
χ CPn ≥ n/2. Therefore, χ CPn n/2.
Remark 3.4. If n 1, 2, 3, 4, then χ CPn 1, 2, 3, 3, respectively.
References
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Providence, RI, USA, 2004.
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