On the uniqueness of d-vertex magic constant
Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant....
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Zielona Góra
2014-05-01
|
| Series: | Discussiones Mathematicae Graph Theory |
| Subjects: | |
| Online Access: | https://doi.org/10.7151/dmgt.1728 |
| _version_ | 1797704376370855936 |
|---|---|
| author | Arumugam S. Kamatchi N. Vijayakumar G.R. |
| author_facet | Arumugam S. Kamatchi N. Vijayakumar G.R. |
| author_sort | Arumugam S. |
| collection | DOAJ |
| description | Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4. |
| first_indexed | 2024-03-12T05:19:28Z |
| format | Article |
| id | doaj.art-27ccbba92939440091a6f52cce08c247 |
| institution | Directory Open Access Journal |
| issn | 2083-5892 |
| language | English |
| last_indexed | 2024-03-12T05:19:28Z |
| publishDate | 2014-05-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae Graph Theory |
| spelling | doaj.art-27ccbba92939440091a6f52cce08c2472023-09-03T07:47:21ZengUniversity of Zielona GóraDiscussiones Mathematicae Graph Theory2083-58922014-05-0134227928610.7151/dmgt.1728dmgt.1728On the uniqueness of d-vertex magic constantArumugam S.0Kamatchi N.1Vijayakumar G.R.2National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126, Tamil Nadu, IndiaNational Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126, Tamil Nadu, IndiaVijayakumar School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Colaba, Mumbai 400 005, IndiaLet G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.https://doi.org/10.7151/dmgt.1728distance magic graphd-vertex magic graphmagic constantdominating functionfractional domination numbe |
| spellingShingle | Arumugam S. Kamatchi N. Vijayakumar G.R. On the uniqueness of d-vertex magic constant Discussiones Mathematicae Graph Theory distance magic graph d-vertex magic graph magic constant dominating function fractional domination numbe |
| title | On the uniqueness of d-vertex magic constant |
| title_full | On the uniqueness of d-vertex magic constant |
| title_fullStr | On the uniqueness of d-vertex magic constant |
| title_full_unstemmed | On the uniqueness of d-vertex magic constant |
| title_short | On the uniqueness of d-vertex magic constant |
| title_sort | on the uniqueness of d vertex magic constant |
| topic | distance magic graph d-vertex magic graph magic constant dominating function fractional domination numbe |
| url | https://doi.org/10.7151/dmgt.1728 |
| work_keys_str_mv | AT arumugams ontheuniquenessofdvertexmagicconstant AT kamatchin ontheuniquenessofdvertexmagicconstant AT vijayakumargr ontheuniquenessofdvertexmagicconstant |