A Note on Cordial Labelings of Multiple Shells
Mahesh M. Andar, Samina Abbas Boxwala, Nirmala B. Limaye · Birkhäuser Basel eBooks · 2002
Let G be a graph with vertex set V and edge set E. A vertex labelling $$f:V \to \left\{ {0,1} \right\}$$ induces an edge labelling $$\bar{f}:E \to \left\{ {0,1} \right\}$$ defined by $$\bar{f}\left( {uv} \right) = \left| {f\left( u \right) - f\left( v \right)} \right|$$ . Let $${{v}_{f}}\left( 0 \right),{{v}_{f}}\left( 1 \right)$$ denote the number of vertices v with f (v)= 0 and f (v) = 1 respectively. Let e f (0), e f (1) be similarly defined. A graph is said to be cordial if there exists a vertex labeling f such that $$\left| {{{v}_{f}}\left( 0 \right) - {{v}_{f}}\left( 1 \right)} \right| \leqslant 1$$ and $$\left| {{{e}_{f}}\left( 0 \right) - {{e}_{f}}\left( 1 \right)} \right| \leqslant 1$$ . In this paper, we show that every multiple shell $$ MS\left\{ {n_{1}^{{{\text{ }}{{t}_{1}}}}, \ldots ,n_{r}^{{{\text{ }}{{t}_{r}}}}} \right\} $$ is cordial for all positive integers $${{n}_{1}}, \ldots ,{{n}_{r}},{{t}_{1}}, \ldots ,{{t}_{r}}$$