J. Comp. & Math. Sci. Vol.2 (6), 836-841 (2011)
Cordial Labeling of Graphs under Barycentric Subdivision M. A. BASKER Department of Mathematics, Loyola College, Chennai-600 034, India ABSTRACT A function f on the vertex set to the set {0, 1}, where each edge xy is assigned the label |f(x) – f(y)| is called a cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1, and the number of edges labeled 0 and the number of edges labeled 1 differ by at most by 1. In this paper we prove that the barycentric subdivision of cycles is 3-cordial. The path union of such cycles is also proved to be 3-cordial. Keywords: Cordial labeling, barycentric subdivision, path union of graphs.
1. INTRODUCTION A labeling of a graph is a function from the vertex set of the graph to the natural numbers. Often the labels of the vertices induce labels of the edges. This can happen in several different ways, but the two most common methods of defining the edge labels are to consider the absolute value of either the difference or the sum of the vertex labels, the sum taken modulo a fixed integer, usually the number of edges of the graph. All graphs considered in this paper are finite, simple, connected and undirected. For a graph G (V , E ) we let p = V and q= E .
A function f is called a graceful labeling of a graph G if f is an injective function from V (G ) to the set {0,1 ... q} , such that when each edge uv is assigned the label f (u) − f (v) , the resulting edge labels are distinct. If a graph G admits a graceful labeling, then we say that G is a graceful graph. The graceful labeling problem is to determine which graphs are graceful. Proving a graph is or is not graceful involves either producing a graceful labeling or showing that no such labeling exists. A function f is called a harmonious labeling of a graph G if f is an injective function from V (G ) to the set
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M. A. Basker, J. Comp. & Math. Sci. Vol.2 (6), 836-841 (2011)
{0,1 ... q − 1}
such that when each edge uv is assigned the label f (u ) + f (v)(mod q) , the resulting edge labels are distinct. If a graph G admits a harmonious labeling, then we say that G is a harmonious graph. Cahit1 has introduced a variation of both graceful and harmonious labelings. Let f be a function from the vertex set of a graph G to {0,1} and let each edge xy be assigned the label f ( x) − f ( y ) . Then f is a cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ at most by 1 and the number of edges labeled 0 and the number of edges labeled 1 differ at most by 1 . Hovey7 has introduced a simultaneous generalization of both harmonious and cordial labelings. For any abelian group A (under addition), a graph G is said to be A cordial if there is a labeling of V with elements of A so that, for all a and b in A , when the edge ab is labeled with f (a) + f (b) , the number of vertices labeled with a and the number of vertices labeled with b differ by at most 1 and the number of edges labeled with a and the number of edges labeled with b differ by at most 1 . In the case where A is cyclic group of order k , the labeling is called k -cordial. With this definition we have: G is harmonious if and only if G is E -cordial; G is cordial if and only if G is 2-cordial. In this paper we identify classes of graphs that are 3-cordial. 2. AN OVERVIEW OF THE PAPER Cahit2 proved the following: every tree is cordial; K n is cordial if and only if
n ≤ 3 ; K m , n is cordial for all m and n ; the
friendship graph C3(t ) (i.e., the one-point union of t 3 -cycles) is cordial if and only if t ≡/ 2(mod 4) ; all fans are cordial; the wheel Wn is cordial if and only if n ≠ 3(mod 4) ; maximal outerplanar graphs are cordial; and an Eulerian graph is not cordial if its size is congruent to 2(mod 4) . Kuo, Chang, and Kwong10 determined all m and n for which mK n is cordial. Seoud et al.12 proved that if G is a graph with n vertices and m edges and every vertex has odd degree then G is not cordial when m + n ≡ 2(mod 4) They have also proved the following: for m ≥ 2 , Cn × Pm is cordial except for the case C4 k + 2 × Pm ; Pn is cordial for all n ( Pnk , k th 2
power of Pn , is the graph obtained from Pn by adding edges that join all vertices u and v with d (u, v) = k ); Pn3 is cordial if and only if n ≠ 4 ; and Pn4 is cordial if and only if 13 n ≠ 4, 5 or 6 . Seoud et al. have proved that the following graphs are cordial: Pn + Pm for all m and n except (m, n) = (2, 2) ; Cm + Cn if m ≡ 0(mod 4) and n ≠ 2(mod 4) ; Cn + K1, m for n ≡/ 3(mod 4) and odd except (n, m) = (3,1) ; C n + K m when n is odd, and when n is even and m is odd. Diab3 proved that the following graphs are cordial: Cm + Pn if and only if (m, n) ≠ (3,3), (3, 2) or (3,1) ; Pm + K1, n if and only if (m, n) ≠ (1, 2) ; Pm ∪ K1, n if and only if (m, n) ≠ (1, 2) ; Cm ∪ K1, n ; C m + K n ; for all m and n except m ≡ 3(mod 4) and n odd, and m ≡ 2(mod 4) and n even; Cm ∪ K n for all
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
M. A. Basker, J. Comp. & Math. Sci. Vol.2 (6), 836-841 (2011)
m and n except m ≡ 2(mod 4) ; Pm + K n ; and Pm ∪ K n . Youssef16 has proved the following: If G and H are cordial and one has even size then G ∪ H is cordial; if G and H are cordial and both have even size then G + H is cordial; if G and H are cordial and one has even size and either one has even order, then G + H is cordial. Lee and Liu11 provide the general construction for the forming of cordial graphs from smaller cordial graphs.
A k -angular cactus is a connected graph all of whose blocks are cycles with k vertices. Cahit2 proved that a k -angular cactus with t cycles is cordial if and only if kt ≅ 2(mod 4) . This was improved by Kirchherr8 who showed that any cactus whose blocks are cycles is cordial if and only if the size of the graph is not congruent to 2(mod 4) . Kirchherr9 also gave a characterization of cordial graphs in terms of their adjacency matrices. Ho et al.6 proved: Pn × C4 m is cordial for all m and all odd n ; the composition G[ H ] is cordial if G is cordial and H is cordial and has odd order and even size. The same authors5 showed that a unicyclic graph is cordial unless it is C4k + 2 and that the generalized Petersen graph P(n, k ) is cordial if and only if 4 n ≅ 2(mod 4) . Du determine the maximum number of edges in a cordial graph of order n and gives a necessary condition for a k regular graph to be cordial. Hovey7 has obtained the following: caterpillars are k -cordial for all k ; all trees are k -cordial for k = 3, 4 and 5 ; odd cycles with pendent edges attached are k -cordial
838
for all k ; cycles are k -cordial for all odd k ; for k even, C2mk + j is k -cordial when 0 ≤ j ≤ k2 + 2 and when k < j < 2k ; C(2 m +1) k
is not k -cordial; K m is 3 -cordial; and, for k -even, K mk is k -cordial if and only if m =1. 3. BARYCENTRIC SUBDIVISION OF CYCLES Creating a barycentric subdivision is a recursive process. In this section we consider the concept of barycentric subdivision of cycles introduced by Vaidya et al.15. We recall this concept now. An edge e = uv of a graph G is said to be subdivided when it is deleted and replaced by path of length 2 . Let Cn = u1 ... u1 be a cycle on n vertices. Subdivide each edge ui ui +1 of Cn and let the new vertex be ui ' , 1 ≤ i ≤ n . Join ui ' with u 'i +1 , 1 ≤ i ≤ n . All suffixes are taken modulo n . The resulting graph is denoted as Cn 2 . This graph is called the barycentric subdivision of 15 Cn and it is like inscribing Cn in Cn . See Figure 1. Vaidya et al.15 have proved that barycentric subdivision of Cn is cordial and path union of finite copies of Cn 2 is also cordial. In this paper we extend the results and prove that Cn 2 and the path union of finite copies of Cn 2 are 3 -cordial. Theorem 1: The graph Cn 2 is 3 -cordial when n ≡ 0(mod 3), n ≥ 6 . Proof Let u1 , u2 ... un be the vertices of the cycle Cn taken in the anticlockwise order. Let u1' , u2' ... un' be the newly inserted vertices
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M. A. Basker, J. Comp. & Math. Sci. Vol.2 (6), 836-841 (2011)
in the barycentric subdivision of Cn where ui' subdivides the edge ui ui +1 . Define a vertex labeling f : V (Cn2 ) → {0,1, 2} by f (ui ) = 0 = f (ui' ) for i ≡ 1(mod 3) f (ui ) = 1 = f (ui' ) for i ≡ 2(mod 3) f (ui ) = 2 = f (ui' ) for i ≡ 0(mod 3)
(1)
The vertex labels given by (1) induce the following labels on edges. Denoting the edge label also by f we have
Let v f (0) , v f (1) and v f (2) be the number of vertices of Cn 2 receiving labels 0 , 1 , and 2 respectively, corresponding to the labeling function f . The definition of f implies that the labels of vertices of Cn 2 follow the sequence pattern 0,0,1,1,2,2,0,0,1,1,2,2,...,0,0,1,1,2,2. Since n ≡ 0(mod 3) and since Cn 2 has 2n vertices, this means that v f (0) = v f (1) = v f (2) = 23n . Consider the cycles C O = u1u1' u2 u2' ... un u n' u1 and C I = u1' u2' ... u n' u1' called the outer and inner cycles of Cn 2 . The induced edge labels of C O follow the pattern 0,1, 2,0,1, 2,..., 0,1, 2 and
those of C I follow 1, 0, 2,1, 0, 2...1, 0, 2 . If e f (0) , e f (1) and e f (2) denote the number of edges of Cn 2 receiving labels 0,1 and 2 respectively then, since Cn 2 has 3n edges, it is clear that e f (0) = e f (1)e f = (2) = n . See Figure 1: The graph C62 and its 3-cordial labeling
Case 1: i ≡ 1(mod 3) f (ui ui' ) = f (ui ) + f (ui' ) = 0 + 0 = 0 f (ui' ui +1 ) = f (ui' ) + f (ui +1 ) = 0 + 1 = 1 f (ui' ui' +1 ) = f (ui' ) + f (ui' +1 ) = 0 + 1 = 1
Case 2: i ≡ 2(mod 3) f (ui ui' ) = f (ui ) + f (ui' ) = 1 + 1 = 2 f (ui' ui +1 ) = f (ui' ) + f (ui +1 ) = 1 + 2 = 0 f
(ui' ui' +1 )
= f
(ui' )
+ f
(ui' +1 )
= 1+ 2 = 0
Case3: i ≡ 0(mod 3) f (ui ui' ) = f (ui ) + f (ui' ) = 2 + 2 = 1 f (ui' ui +1 ) = f (ui' ) + f (ui +1 ) = 2 + 0 = 2 f (ui' ui' +1 ) = f (ui' ) + f (ui' +1 ) = 2 + 0 = 2
Figure 1. Inscribing Cn in Cn , k number of times we obtain a graph which we denote by Cnk . Theorem 2: The graph Cnk , k > 2 is 3cordial when n ≡ 0(mod 3) , n ≥ 6 . The proof of Theorems 2 is similar to that of Theorem 1. 4. PATH UNION OF GRAPHS Let G1 , G2 ...Gn , n ≥ 2 , be copies of a fixed graph G . Shee and Ho14 call a graph obtained by adding an edge from Gi to Gi +1 for i = 1, 2 ... n − 1 , a path-union of G . Among their results they show the following
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
M. A. Basker, J. Comp. & Math. Sci. Vol.2 (6), 836-841 (2011)
graphs are cordial: path-unions of cycles; path-unions of n copies of K m when m = 4, 6 or 7 . Let Pm be a path on m vertices. Consider m copies of Cn 2 and identify each vertex of Pm with a 2 -degree vertex of each copy of Cn 2 . In this paper we call this graph as a path union m copies of Cn 2 . Theorem 3: The path union of m copies of C n 2 is 3 -cordial. Proof: Let v1 , v2 ... vm be the vertices of path Pm . Let G be the path union of m copies of C n 2 and let G1 , G2 ... Gm be m copies of C n 2 . Hovey7 has proved that all trees are k -
840
cordial for k = 3, 4 and 5 . This means that a path is 3 -cordial. Let f denote a 3 -cordial labeling of Pm defined by 0 if i ≡ 1(mod 3) f (vi ) = 1 if i ≡ 2(mod 3) 2 if i ≡ 0(mod 3)
Identify the vertex vi with a 2 degree vertex ui of Gi if and only if vi and ui receive the same label in the respective 3 -cordial labeling of Pm and C n 2 .See Figure 2. Thus the path union is 3 -cordial.
Figure 2 : Path union of 3 copies of C62
Theorem 4: The path union of m copies of Cnk is 3 -cordial.
6. REFERENCES 1.
5. CONCLUSION In this paper we have proved that barycentric subdivision of Cnk and path union of Cnk are 3 -cordial, for k ≥ 2 and n ≡ 0(mod 3), n ≥ 6 . Identifying classes of graphs which are 3-cordial under barycentric subdivision is open.
2.
3.
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