Even Vertex Equitable Even Labeling for Cycle Related Graphs

A. Lourdusamy ; F. Patrick

Kragujevac Journal of Mathematics, Tome 43 (2019) no. 3, p. 427

Résumé

Let $G$ be a graph with $p$ vertices and $q$ edges and $A=\{0,2,4,\ldots,q+1\}$ if $q$ is odd or $A=\{0,2,4,\ldots,q\}$ if $q$ is even. A graph $G$ is said to be an even vertex equitable even labeling if there exists a vertex labeling $f:V(G) \rightarrow A$ that induces an edge labeling $f^{*}$ defined by $f^{*}(uv)=f(u)+f(v)$ for all edges $uv$ such that for all $a$ and $b$ in $A$, $|v_{f}(a)-v_{f}(b)| \leq 1$ and the induced edge labels are $2,4,\ldots,2q$, where $v_{f}(a)$ be the number of vertices $v$ with $f(v)=a$ for $a \in A$. A graph that admits even vertex equitable even labeling is called an even vertex equitable even graph. In this paper, we prove that the graphs $C_{m} \ominus P_{n}$, $C_{n}(Q_{m})$ if $n \equiv 0,3 \pmod 4$, $\left[ P_{n};C^{(2)}_{m} \right]$ if $m \equiv 0\pmod 4$, $C_{m} *_{e} C_{n}$ and the graph obtained by duplicating an arbitrary vertex and edge of a cycle $C_{n}$ admit an even vertex equitable even labeling.

Détail

Classification : 05C78
Keywords: Vertex equitable labeling, even vertex equitable even labeling