On Divisor 3-Equitable Labeling of Wheel Graphs
K. Tina Jebi Nivathitha1, N. Srinivasan2, A. Parthiban3, Sangeeta4
1K. Tina Jebi Nivathitha, Research Scholar, Department of Mathematics, St. Peter’s Institute of Higher Education and Research, Avadi, Chennai-600 054, Tamil Nadu, India.
2Dr. N. Srinivasan, Professor and Head, Department of Mathematics, St. Peter’s Institute of Higher Education and Research, Avadi, Chennai-600 054, Tamil Nadu, India.
3Dr. A. Parthiban*, Assistant Professor, Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Phagwara-144 411, Punjab, India.
4Mrs. Sangeeta, Research Scholar, Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Phagwara-144 411, Punjab, India.
Manuscript received on January 02, 2020. | Revised Manuscript received on January 15, 2020. | Manuscript published on January 30, 2020. | PP: 5550-5553 | Volume-8 Issue-5, January 2020. | Retrieval Number: 10.35940/ijrte.E7040.018520 | DOI: 10.35940/ijrte.E7040.018520
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Abstract: A graph 𝑮 on 𝒏 vertices is said to admit a divisor 3-equitable labeling if there exists a bijection 𝒅∶𝑽(𝑮)→ {𝟏,𝟐,…,𝒏} defined by 𝒅 𝒆=𝒙𝒚 = 𝟏,𝒊𝒇𝒅(𝒙)|𝒅 𝒚 or 𝒅 𝒚 |𝒅(𝒙)𝟐, 𝒊𝒇 𝒅 𝒙 𝒅 𝒚 =𝟐 𝒐𝒓𝒅 𝒚 𝒅 𝒙 =𝟐𝟎, 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆 and |𝒆𝒅 𝒊 −𝒆𝒅 𝒋 |≤ 𝟏 for all 𝟎 ≤ 𝒊,𝒋≤ 𝟐, where 𝒆𝒅 𝒊 denotes the number of edges labelled with “𝒊”. A graph which permits a divisor 3-equitable labeling is called a divisor 3-equitable graph. A wheel graph 𝑾𝒏 is defined as 𝑾𝒏=𝑪𝒏−𝟏⋀ 𝑲𝟏, where 𝑪𝒏−𝟏 is a cycle on 𝒏−𝟏 vertices and 𝑲𝟏 is a complete graph on a single vertex. In this paper, we prove the non-existence of a divisor 3-equitable labeling of the wheel graph 𝑾𝒏 for 𝒏≥𝟕.
Keywords:
Keywords: 3-equitable labeling, Divisor 3-equitable labeling, Wheel graph.
Scope of the Article: Graph Algorithms And Graph Drawing