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 \(C_n(v_1, v_2, \ldots, v_{n-1})\) 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 \(C_{2n}(v_0,v_1, v_2, \ldots, v_{2n-1})\) by adding \(n-2\) chords between the vertex \(v_0\) and the vertices \(v_{2i+1}\), for \(1\leq i \leq n-2\). 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.