CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI Volume 4 (4) (2017), Pages 161-166 p-ISSN: 2086-0382; e-ISSN: 2477-3344
Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs
Fery Firmansah1, Muhammad Ridlo Yuwono2
1, 2Mathematics Edu. Depart. University of Widya Dharma Klaten, Indonesia
Email: [email protected], [email protected]
ABSTRACT A graph 퐺(푉(퐺), 퐸(퐺)) is called graph 퐺(푝, 푞) if it has 푝 = |푉(퐺)| vertices and 푞 = |퐸(퐺)| edges. The graph 퐺(푝, 푞) is said to be odd harmonious if there exist an injection 푓: 푉(퐺) → {0,1,2, … ,2푞 − 1} such that the induced function 푓∗: 퐸(퐺) → {1,3,5, … ,2푞 − 1} defined by 푓∗(푢푣) = 푓(푢) + 푓(푣). The function 푓∗ is a bijection and 푓 is said to be odd harmonious labeling of 퐺(푝, 푞). In this paper we prove that pleated of the (푘) Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 are odd harmonious graph. Moreover, we also give odd (푘) (푘) harmonious labeling construction for the union pleated of the Dutch windmill graph 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1.
Keywords: odd harmonious labeling, pleated graph, the Dutch windmill graph
INTRODUCTION In this paper we consider simple, finite, connected and undirected graph. A graph 퐺(푝, 푞) with 푝 = |푉(퐺)| vertices and 푞 = |퐸(퐺)| edges. A graph labeling which has often been motivated by practical problems is one of fascinating areas of research. Labeled graphs serves as useful mathematical models for many applications in coding theory, communication networks, and mobile telecommunication system. We refer to Gallian [1] for a dynamic survey of various graph labeling problems along with extensive bibliography. Most graph labeling methods trace their origin to one introduced by Rosa in 1967, or one given by Graham and Sloane in 1980.
Graham and Sloane [2] introduced and defined harmonious labeling as follows: Definition 1. A graph 퐺(푝, 푞) is said to be harmonious if there is an exist injection 푓: 푉(퐺) → 푍푞 ∗ ∗ such that the induced function 푓 : 퐸(퐺) → 푍푞 defined by 푓 (푢푣) = (푓(푢) + 푓(푣))(푚표푑 푞) is a bijection and 푓 is said to be harmonious labeling of 퐺(푝, 푞).
Liang and Bai [3] introduced and defined odd harmonious labeling as follows: Definition 2. A graph 퐺(푝, 푞) is said to be odd harmonious if there is an exist injection 푓: 푉(퐺) → {0,1,2, … ,2푞 − 1} such that the induced function 푓∗: 퐸(퐺) → {1,3,5, … ,2푞 − 1} defined by 푓∗(푢푣) = 푓(푢) + 푓(푣) is a bijection and 푓 is said to be odd harmonious labeling of 퐺(푝, 푞).
A graph that admits odd harmonious labeling is called odd harmonious graphs. Liang and Bai [3] have obtained the necessary conditions for the existence of odd harmonious labeling graph. They proved if 퐺 is an odd harmonious graph, then 퐺 is a bipartite graph and 퐺(푝, 푞) is an
Submitted: 7 Pebruary 2017 Reviewed: 24 March 2017 Accepted: 19 May 2017 DOI: http://dx.doi.org/10.18860/ca.v4i4.4043
Odd Harmonious Labeling on Pleated of the Dutch Windmil Graphs
odd harmonious labeling then number vertices is bounded by √푞 ≤ 푝 ≤ 2푞 − 1. The maximal label of all vertices in an odd harmonious graph 퐺 is at most 2푞 − 훿(퐺), where 훿(퐺) is the minimum degree of the vertices of 퐺. In the same paper Liang and Bai [3] proved that 퐶푛 is odd harmonious if and only if 푛 ≡ 0 (푚표푑 4) and a complete graph 퐾푛 is odd harmonious if and only if 푛 = 2. The odd harmoniousness of graph is useful for the solution of undetermined equations. Alyani, Firmansah, Giyarti and Sugeng [4] proved the odd harmonious labeling of 푘퐶푛 snake graphs for specific values of 푛, that is, for 푛 = 4 and 푛 = 8. Firmansah [5] proved that union of snake graph 푘퐶4 ∪ 푘퐶4 with 푘 ≥ 1 and pleated of snake graph 푘퐶4(푟) with 푘 ≥ 1 and 푟 ≥ 1 admitted odd harmonious graph. Firmansah and Sugeng [6] proved that the Dutch wind mill graph (푘) (푘) (푘) 퐶4 with 푘 ≥ 1 and union of the ducth windmill graph 퐶4 ∪ 퐶4 with 푘 ≥ 1 admited odd harmonious labeling. Firmansah [7] proved that the quadrilateral windmill graph 퐷푄(푘) with 푘 ≥ 1 admitted odd harmonious graph. Several results have been published on odd harmonious labeling see [8], [9], [1], [10], [11], [12], [13], and [14]. (푘) In this paper, we prove that pleated of the Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 are the odd harmonious graph. Moreover, we also give the odd harmonious labeling (푘) (푘) construction for the union pleated of the Dutch windmill graph 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1.
(푘) Definition 3. [6] The Dutch windmill graphs 퐶4 with 푘 ≥ 1 is a graph which consists of 푘 copies of cycle graphs 퐶4 with a common central vertex 푢0.
Definition 4. [5] The pleated cycle graphs 퐶4(푟) with 푟 ≥ 1 is a graph formed from cycle graphs 퐶4 with vertex set {푢0, 푣1, 푣2, 푢1} by adding {푢2, 푢3, … , 푢푟} vertices which are connected with vertex 푣1 and 푣2.
RESULTS AND DISCUSSIONS We have some observation to show the result of odd harmonious graph. (푘) Definition 5. The pleated of the Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is a graph which consists of 푘 copies of the pleated cycle graphs 퐶4(푟) with a common central vertex 푢0.
The vertex notation and construction of the pleated cycle graph 퐶4(푟) with 푟 ≥ 1 and (푘) pleated of the Dutch windmill graph 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is shown in Figure 1. r u1
2 u1
ur 1 2 2v1 v1 1 v 1 v r k u1 2 r u k 2 2 u 2 1 u uk 0 1 u2 uk u 2 u2 1 2 vk v2 u v1 1 v2
u0 C r C (k ) r 4 4 (푘) Figure 1. The pleated cycle graph 퐶4(푟) and pleated of the Dutch windmill graph 퐶4 (푟)
Fery Firmansah 162 Odd Harmonious Labeling on Pleated of the Dutch Windmil Graphs
(푘) (푘) Definition 6. A graphs 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is a union pleated of the Dutch (푘) windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1.
The vertex notation and construction the union pleated of the Dutch windmill graphs (푘) (푘) 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is shown in Figure 2.
u r r 1 x1
2 2 u1 x1
1 2 1 2 2v1 v1 1 2 y1 y1 1 1 1 r vk u v2 yk y2 u 1 r r x1 r k 2 2 u 2 xk 2 2 x 2 u 1 u0 1 u 1 x0 x k u u 2 xk x 1 2 k 2 k x 2 1 2 1 2 vk v2 yk y2
C ( k ) r C ( k ) r 4 4
(푘) (푘) Figure 2. The union pleated of the Dutch windmill graphs 퐶4 (푟) ∪ 퐶4 (푟)
(푘) Theorem 1. The pleated of the Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is an odd harmonious graph. Proof. (푘) Let G be the pleated of the Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1. (푘) The vertex set and edge set of 퐶4 (푟) are defined as follows 푗 푚 푉(퐺) = {푢0} ∪ {푣푖 |1 ≤ 푖 ≤ 푘, 푗 = 1,2} ∪ {푢푖 |1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟} and 푗 푗 푚 퐸(퐺) = {푢0푣푖 |1 ≤ 푖 ≤ 푘, 푗 = 1,2} ∪ {푣푖 푢푖 |1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟, 푗 = 1,2} then 푝 = |푉(퐺)| = 푘푟 + 2푘 + 1 and 푞 = |퐸(퐺)| = 2푘푟 + 2푘. Define the vertices labels 푓: 푉(퐺) → {0,1,2,3 … ,4푘푟 + 4푘 − 1} as follows 푓(푢0) = 0 푗 푓(푣푖 ) = 4푖 + 2푗 − 5, 1 ≤ 푖 ≤ 푘, 푗 = 1,2 푚 푓(푢푖 ) = (4푟 + 4)푘 − (4푟 + 4)푖 + 4푚, 1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟 The labeling 푓 will induce the mapping 푓∗: 퐸(퐺) → {1,3,5,7, … ,4푘푟 + 4푘 − 1} which is defined by 푓∗(푢푣) = 푓(푢) + 푓(푣). Thus we have the edges labels as follows ∗ 푗 푓 (푢0푣푖 ) = 4푖 + 2푗 − 5, 1 ≤ 푖 ≤ 푘, 푗 = 1,2 ∗ 푗 푚 푓 (푣푖 푢푖 ) = (4푟 + 4)푘 − 4푟푖 + 2푗 + 4푚 − 5, 1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟, 푗 = 1,2 It is not difficult to show that the mapping 푓 is an injective mapping and the mapping 푓∗ admits a (푘) bijective mapping. Hence pleated of the Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is an odd harmonious graph. ∎
Fery Firmansah 163 Odd Harmonious Labeling on Pleated of the Dutch Windmil Graphs
(푘) (푘) Theorem 2. The union pleated of the Dutch windmill graphs 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is an odd harmonious graph. Proof. (푘) (푘) Let 퐺 be the union pleated of the Dutch windmill graphs 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1. The vertex set and edge set of 퐺 are defined as follows 푗 푚 푉(퐺) = {푢0} ∪ {푣푖 |1 ≤ 푖 ≤ 푘, 푗 = 1,2} ∪ {푢푖 |1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟} ∪ {푥0} ∪ 푗 푚 {푦푖 |1 ≤ 푖 ≤ 푘, 푗 = 1,2} ∪ {푥푖 |1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟} and 푗 푗 푚 퐸(퐺) = {푢0푣푖 |1 ≤ 푖 ≤ 푘, 푗 = 1,2} ∪ {푣푖 푢푖 |1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟, 푗 = 1,2} ∪ 푗 푗 푚 {푥0푦푖 |1 ≤ 푖 ≤ 푘, 푗 = 1,2} ∪ {푦푖 푥푖 |1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟, 푗 = 1,2} then 푝 = |푉(퐺)| = 2푘푟 + 4푘 + 2 and 푞 = |퐸(퐺)| = 4푘푟 + 4푘. Define the vertices labels 푓: 푉(퐺) → {0,1,2,3 … ,8푘푟 + 8푘 − 1} as follows 푓(푢0) = 0 푗 푓(푣푖 ) = 4푖 + 2푗 − 5, 1 ≤ 푖 ≤ 푘, 푗 = 1,2 푚 푓(푢푖 ) = (4푟 + 4)푘 − (4푟 + 4)푖 + 4푚, 1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟 푓(푥0) = 2 푗 푓(푦푖 ) = (4푟 + 4)푘 + 4푖 + 2푗 − 7, 1 ≤ 푖 ≤ 푘, 푗 = 1,2 푚 푓(푥푖 ) = (4푟 + 4)푘 − (4푟 + 4)푖 + 6푚, 1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟 The labeling 푓 will induce the mapping 푓∗: 퐸(퐺) → {1,3,5,7, … ,8푘푟 + 8푘 − 1} which is defined by 푓∗(푢푣) = 푓(푢) + 푓(푣). Thus we have the edges labels as follows ∗ 푗 푓 (푢0푣푖 ) = 4푖 + 2푗 − 5, 1 ≤ 푖 ≤ 푘, 푗 = 1,2 ∗ 푗 푚 푓 (푣푖 푢푖 ) = (4푟 + 4)푘 − 4푟푖 + 2푗 + 4푚 − 5, 1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟, 푗 = 1,2 ∗ 푗 푓 (푥0푦푖 ) = (4푟 + 4)푘 + 4푖 + 2푗 − 5, 1 ≤ 푖 ≤ 푘, 푗 = 1,2 ∗ 푗 푚 푓 (푦푖 푥푖 ) = (8푟 + 8)푘 − 4푟푖 + 2푗 + 6푚 − 7, 1 ≤ 푖 ≤ 푘, 1 ≤ 푚 ≤ 푟, 푗 = 1,2 It is not difficult to show that the mapping 푓 is an injective mapping and the mapping 푓∗ admits a (푘) (푘) bijective mapping. Hence union pleated of the Dutch windmill graphs 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 is an odd harmonious graph. ∎
(5) Example 1. Odd harmonious labeling for pleated of the Dutch windmill graph 퐶4 (4) is shown in Figure 3. 96
92
88
84
16 1 3 76 12 72 8 19 5 68 4 64 0 17 7
15 9
13 11 24 44
28 48
32 52
36 56
(5) Figure 3. Pleated of the Dutch windmill graph 퐶4 (4) and its odd harmonious labeling.
Fery Firmansah 164 Odd Harmonious Labeling on Pleated of the Dutch Windmil Graphs
Example 2. Odd harmonious labeling for the union pleated of the Dutch windmill graphs (4) (4) (5) (5) 퐶4 (3) ∪ 퐶4 (3) and 퐶4 (4) ∪ 퐶4 (4) is shown in Figure 4.
60 62
56 58
52 54
1 3 63 65
15 5 77 67 0 2 12 8 4 36 40 44 14 10 6 38 42 46 13 7 75 69
11 9 73 71
20 22
24 26
28 30 C (4) 3 C (4) 3 4 4
96 98
92 94
88 90
84 86
16 76 18 1 3 99 101 78 12 72 14 74 19 5 8 68 10 117 103 70 4 64 6 66 0 2 17 7 115 105
15 9 113 107
13 11 111 109 24 44 26 46
28 48 30 50
32 52 34 54
36 56 38 58 C 5 4 C 5 4 4 4 (4) (4) Figure 4. The union pleated of the Dutch windmill graph 퐶4 (3) ∪ 퐶4 (3), (5) (5) 퐶4 (4) ∪ 퐶4 (4) and its odd harmonious labeling.
CONCLUSION We have shown that the pleated of the Dutch windmill graphs and the union pleated of the Dutch windmill graphs admitted odd harmonious labeling. We suggest that there is an open problem for further research as follows.
Fery Firmansah 165 Odd Harmonious Labeling on Pleated of the Dutch Windmil Graphs
Open Problem. Find if there exists odd harmonious labeling on pleated of the Dutch windmill graph (푘) 퐶푛 (푟) with 푛 ≡ 0 (푚표푑 4), 푘 ≥ 1 and 푟 ≥ 1.
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