On Graceful And Cordial Labeling of Shell Graphs

Abstract

We recall from [13] a shell graph of size $n$, denoted $C(m,n-3)$, is the graph obtained from the cycle $C_n(v_0,v_1,v_2\dots ,v_{n-1})$ by adding $m-3$ consecutive chords incident at a common vertex, say $v_0$. The vertex $v_0$ of $C(n,n-3)$ is called the apex of the shell $C(n,n-3)$. The vertex $v_0$ of $C(n,n-3)$ is said to be at level $l$.

A graph $C(2n,n-2)$ is called an alternate shell, if $C(2n,n-2)$ is obtained from the cycle $C2n(v_0,v_1,v_2\dots ,v_{2n-1})$ by adding $n-2$ chords between the vertex $v_0$ and the vertices $v_{2i-1}$ for $1-i\delta n$. If the vertex $v_i$ of $C(2n,n-2)$ at level $l$ and is adjacent with $v_0$, then $v_l$ is said to be at level $l$ with a chord, otherwise the vertex $v_i$ is said to be at level $l$ without a chord.

A graph, denoted $G2n_i,n_i,2,k,l$, is called one vertex union of alternate shells with a path at any common level $l$ (with or without chords), if it is obtained from $k$ alternate shells $C(2n_i,n_i-2{)}^{\prime }s$, $1-i\delta k$, by merging them together at their apex and joining $k$ vertices each chosen from a distinct alternate shell in a particular level $l$ (with or without chords) by a path $P_{2k-1}$, such that the chosen vertex of the $i$th alternate shell $C(2n_i,n_i-2)$ is at the $(2i-1)$th vertex of the $P_{2k-l}$ for $1-i\delta k$. We denote the graph $G2n_i,n_i,2,k,l$ as $G2n_i,n_i,2,k,l_c$ if the path $P_{2k-1}$ joins the vertices only at the common level $l$ with chords.

In this paper, we show that $G2n_i,n_i,2,k,l_c$ is graceful and admits an $A$-labeling, for $k-\tau 1,n_i$, $3,1\tau 1,n_i$, and $G2n_i,n_i,2,k,1$ is cordial, for $n_i-n-3,k-1,1\tau i$.