On Graceful and Cordial Labeling of Shell Graphs

Abstract

We recall from [13] a shell graph of size $n$, denoted $C(n,n-3)$, is the graph obtained from the cycle $C_n(v_1,v_2,\dots ,v_{n-1})$ by adding $n-3$ 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 $v_1$ of $C(n,n-3)$ is said to be at level 1.

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