New results on vertex equitable labeling

Abstract

The concept of vertex equitable labeling was introduced in [9]. A graph $G$ is said to be vertex equitable if there exists a vertex labeling $f$ such that for all $a$ and $b$ in $A$, $\left|v_f(a)-v_f(b)\right|\leq1$ and the induced edge labels are $1, 2, 3,\cdots, q$. A graph $G$ is said to be a vertex equitable if it admits a vertex equitable labeling. In this paper, we prove that the graphs, subdivision of double triangular snake $S(D(T_n))$, subdivision of double quadrilateral snake $S(D(Q_n))$, subdivision of double alternate triangular snake $S(DA(T_n))$, subdivision of double alternate quadrilateral snake $S(DA(Q_n))$, $DA(Q_m)\odot nK_1$ and $DA(T_m)\odot nK_1$ admit vertex equitable labeling.

Keywords

• Vertex equitable labeling,Vertex equitable graph,Double triangular snake graph,Double alternate triangular snake graph,Double alternate quadrilateral snake graph

Kaynakça

1. [1] J. A. Gallian, Graph labeling, Electron. J. Combin. (2015) (Dynamic Survey #DS6).
2. [2] F. Harary, Graph theory, Addison-Wesley, Reading Mass, 1972.
3. [3] P. Jeyanthi, A. Maheswari, Some results on vertex equitable labeling, Open J. Discrete Math. 2(2) (2012) 51–57.
4. [4] P. Jeyanthi, A. Maheswari, Vertex equitable labeling of transformed trees, J. Algorithms Comput. 44(1) (2013) 9–20.
5. [5] P. Jeyanthi, A. Maheswari, Vertex equitable labeling of cyclic snakes and bistar graphs, J. Sci. Res. 6(1) (2014) 79–85.
6. [6] P. Jeyanthi, A. Maheswari, M. Vijayalaksmi, Vertex equitable labeling of cycle and star related graphs, J. Sci. Res. 7(3) (2015) 33–42.
7. [7] P. Jeyanthi, A. Maheswari, Vertex equitable labeling of cycle and path related graphs, Util. Math. 98 (2015) 215–226.
8. [8] P. Jeyanthi, A. Maheswari, M. Vijayalakshmi, Vertex equitable labeling of double alternate snake graphs, J. Algorithms Comput. 46 (2015) 27–34.

9. [9] M. Seenivasan, A. Lourdusamy, Vertex equitable labeling of graphs, J. Discrete Math. Sci. Cryptogr. 11(6) (2008) 727–735.