eu EVEN SEQUENTIAL HARMONIOUS LABELING OF SOME CYCLE RELATED GRAPHS

In this paper, we generalize this result on cycles by showing that the kCn snake with string 1, 1, · · · , 1 when n ≡ 0(mod4) are even sequential harmonious graph. Also we show that the kC4 snake with m-pendant edges for each k,m ≥ 1, (for linear case and for general case). Moreover, we show that, all subdivision of 2mk snake are even sequential harmonious for each k,m ≥ 1 . Finally we present some examples to illustrate the proposed theories. AMS Subject Classification: 05C78


Introduction
Graph labeling is an active area of research in graph theory which has mainly Received: December 19, 2013 c 2014 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author evolved through its many applications in coding theory, communication networks, mobile telecommunication system.Optimal circuits layouts or graph decompositions problems, no name just a few of them.
Most graph labeling methods trace their origin to one introduced by Rosa [1] called such a labeling a β-valuation and Golomb subsequently called graceful labeling, and one introduced by Graham and Sloane [2] called harmonious labeling.Several infinite families of graceful and harmonious graphs have been reported.Many illustrious works on graceful graphs brought a tide to different ways of labeling the elements of graph such as odd graceful.
All graphs in this paper are finite, simple graphs and undirected graph.The symbols V (G) and E(G) denote the vertex set and edge set of a graph G.The cardinality of the vertex set is called the order of G.The cardinality of the edge set is called the size of G.A graph with p vertices and q edges is called a G(p, q) graph [4].
A graph G of size q is odd-graceful, if there is an injection f : V (G) → {0, 1, 2, , 2q − 1} such that, when each edge xy is assigned the label or weight f (x) − f (y), the resulting edge labels are {1, 3, 5, • • • , 2q − 1}.This definition was introduced by Gnanajothi.Many researchers have studied odd graceful labeling.Seoud and Abdel-Aal determine all connected odd graceful graphs of order less than or equal to 6.For a dynamic survey of various graph labeling problems we refer to Gallian [3].A graph G is said to be odd harmonious if there exists an injection f : Then f is said to be an odd harmonious labeling of G. [8].A graph G with p vertices and q edges is said to have an even sequential harmonious labeling if there exists a function f : then the resulting edge labels are distinct.A graph G is said to be an even sequential harmonious graph if it admits an even sequential harmonious labeling.
Consider a Graph kC n -snake with k ≥ 2. Let u 1 , u 2 , • • • , u k−1 be the consecutive cut vertices of G. Let d i be the distance between u i and u i+1 in G,and the string ( Hence, any graph G = kC n -snake, can be represented by a string.For instance the string (from left to right) of the 5C 4 -snake of figure 4 is 2,2,2,2,2.Gracefulness of the kind of kC 4 -snake studied by Gnanajothi have string 2, 2,.. 2. We obtain in theorem 2.4 an even sequential harmonious labeling of the snake 1, 1 ⌋, we say that snake kC n is linear.
This paper can be divided into two sections.Section 1, we show that the graphs kC 4 -snake ⊙mk 1 (the graph obtained by joining m pendant edges to each vertex of kC 4 -snake k, m ≥ 1) for linear and general cases of kC 4 -snake for each k ≥ 1 are even sequential harmonious.We also obtain an even harmonious labeling of kC n with the sequences string is 1, 1, • • • , 1 and when n ≡ 0(mod4) in section 2, we show that an even sequential harmonious labeling of the all subdivision of of double triangular snakes finally, we prove that the all Subdivision of 2m△ k snake are even sequential harmonious for each k, m > 1.
Proof.Consider the linear graph G = kC 4 -snake, k ≥ 1 which has the vertices w i , u j and v j where i = 0, 1, 2, • • • , k, j = 1, 2, 3, • • • , k.In order to get the linear kC 4 -snake ⊙mk 1 , k, m ≥ 1, we add m pendant edges w l i , u l j andv l j to each vertex of w i , u j , v j respectively such that l = 0, 1, 2, • • • , m.Now, let G be the linear kC 4 -snake ⊙mk 1 , k, m ≥ 1 be described as indicated in figure 1.
The remaining odd edge labels from 8k + l to 2k + 2m(3k + l) are obtained from the following: Hence the graph G is an even sequential harmonious.
Example 2.2.An even sequential harmonious labeling of the graph 3C 4snake ⊙2k 1 is shown in figure 2.
Theorem 2.3.The following graphs are an even sequential harmonious.
Proof.The graph kC 4 -snake can be considered as a bipartite graph (one partite set has black vertices and the other has white vertices) it is possible to embed it, on a square grid as is showed in the figure 3.
Let G be the graph kC 4 -snake ⊙mk 1 which obtained by joining m-pendant edges to each vertex of kC 4 -snake.Then | V (G) |= (m + 1)(3k + 1) and | E(G) = 4k + m(3k + 1) |.Now, we are running the following steps sequentially in order to prove the Theorem ; Step 1.Since kC 4 -snake is a bipartite graph so it has one partite set has black vertices and the other has white vertices as is showed in figure 3. Put black vertices in string ordered by diagonals from left to right and inside each diagonal from bottom to top, assign to them from an arithmetic progression of difference 4 which first term is zero, when we move to another diagonal we use an arithmetic progression of difference 2, counting until the last black vertex has been numbered.Similarly, put the White vertices on a string ordered for diagonals from left to right and inside each diagonal from top to bottom, starting with the first diagonal assign numbers from an arithmetic progression of difference 2 which first term is 1, when we move to another diagonal we use an arithmetic progression of differences 2 counting until the last white vertex has been numbered.
Step 2. In this step, We are labeling the vertices of m-pendant edges which contact with the white diagonals, from the right to left and inside each white diagonal from bottom to top, assign to them from an arithmetic progression of difference 2, which first term is z such that z = y + 2 where y is the last vertex labeling of black diagonal.when we move to a new vertex of the white diagonal, the first vertex of m-pendant edges is labeled by an arithmetic progression of difference 4, but the arithmetic progression of difference 2 has been used with the remain (m − l) vertices of m-pendant edges.We move from a vertex to another of the white diagonals until the last white vertex.
Step 3.Finally we are labeling the vertices of m-pendant edges which contact with the black diagonals, from left to right and inside each black diagonal from bottom to top, assign to them from an arithmetic progression of difference (-2), which first term is (2q − 1) where q is the size of G, When to move to a new vertex of the black diagonal, the first vertex of m-pendant edges is labeled by an arithmetic progression of difference (-6) but the arithmetics progression of difference (-2) has been used with the remain (m − l) vertices of m-pendant edges.We move from a vertex to another of the black diagonal until the last black vertex.Now, we have complete the proof by running the above steps, we mention only the vertices labels and the reader can fulfill the proof as we did in the previous theorem where step 1 give us an even sequential harmonious labeling of the graph kC 4 -snake for each k ≥ 1, and step 1-step 3 give us an even sequential harmonious labeling of the graph kC 4 -snake ⊙mk 1 for each k, m ≥ 1 (in general case).
The following example illustrates the last result.
Example 2.4.An even sequential harmonious labeling of the graph 5C 4snake ⊙2k 1 (for general case) is shown in Figure 4.
It is clear that |E(G)| = 4m(n − 1).We define the labeling function For labeling the vertices v j i1 , 2 ≤ i ≤ n, 2 ≤ j ≤ 2m.We consider the following two cases: Case i: If i is odd, 3 ≤ i ≤ n, we have the following labeling,for each 2 Case ii: If i is even, 2 ≤ i ≤ n, we have the following labeling,for each 2 ≤ j ≤ 2m: It follows that f admits an even sequential harmonious labeling for (n − 1)C 4 m -snake.Hence (n − 1)C 4 m -snake for each n ≥ 2, m ≥ 1 with the string (  The graph 4C8 -snake with its even sequential harmonious labeling

Subdivision of Double Triangles Snake
Rosa [8] defined a triangular snake (or △ -snake) as a connected graph in which all blocks are triangles and the block cut point graph is a path.Let △ k -snake be a △-snake with k blocks while n△ k -snake be a △-snake with k blocks and every block has n number of triangles with one common edge.
David moulton [9] proved that triangular snakes with p triangles are graceful if p is congruent to 0 or 1 modulo 4. Xu [10] proved that they are harmonious if and only if p is congruent to 0 ,1 or 3 modulo 4. A double triangular snake is a graph that formed by two triangular snakes have a common path.The harmonious labeling of double triangle snake introduced by Xi Yue et al [6].It is known that, the graphs which contain odd cycles are not even sequential harmonious so we used the subdivision notation for odd cycle in order to prove that all subdivision of double triangular snakes are even sequential harmonious.Theorem 3.1.All subdivision of double triangular snakes (2△ k -snake, k ≥ 1) are even sequential harmonious.
Proof.Let G = 2△ k -snake has q edges and p vertices.The graph G consists of the vertices Therefore we get the subdivision of double triangular snakes S(G) by subdividing every edge of double triangular 2△ k -snake exactly once.Let y i be the newly added vertex between u i and u i+1 while w i1 and w i2 are newly added vertices between w i u i and w i u i+1 respectively where 1 ≤ i ≤ k.Finally v i1 and v i2 are newly added vertices between v i u i and v i u i+1 respectively, such that 1 ≤ i ≤ k .Let the graph S(G) as described as indicated in Figure 7.It is clear that the number of edges of the graph S(G) is 10k.We define the labeling function : The edge labels will be as follows: • The vertices u i and y i , 1 ≤ i ≤ k, induce the edge labels • The vertices y i and u i+1 , 1 ≤ i ≤ k, induce the edge labels {20i − 10, 1 • The vertices w i1 and w i , So we obtain all the edge labels {2, 4, 6, • • • , 20k}.Hence the subdivision of double triangular snakes (2△ k -snake, k ≥ 1) are even sequential harmonious.Theorem 3.2.All subdivision of 2m△ k -snake, m, k ≥ 1 are even sequential harmonious.
Proof.Let G = 2m△ k -snake has q edges and p vertices.The graph G consists of the vertices (u Therefore we get the generalised subdivision of double triangular snakes S(G) by subdividing every edge of double triangular 2m△ k -snake exactly once.Let y i be the newly added vertex between u i and u i+1 while w j i1 and w j i2 are newly added vertices between u i w j i and w j i u i+1 respectively where 1 ≤ i ≤ k.Finally v j i1 and v j i2 are newly added vertices between u i v j i and v j i u i+1 respectively, such that 1 ≤ i ≤ k, 1 ≤ j ≤ m .Let the graph S(G) as described as indicated in Figure 8.
We define the labeling function f :  In a view of the above defined labeling pattern f is even sequential harmonious for the graph S(G).Hence S(2m△ k -snake) is even sequential harmonious for all m ≥ 1, k ≥ 1.
Illustration 3.3.An even sequential harmonious labeling of the graph subdivision of 6△ 3 -snake is shown in Figure 9.

Figure 4 :
Figure 4: The graph 5C4 -snake ⊙2k1 (for general case) with its even sequential harmonious labeling