We recall from [13] a shell graph of size \(n\), denoted \(C(m,n-3)\), is the graph obtained from the cycle \(C_n(v_0,v_1,v_2\ldots,v_{n-1})\) 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 \(C{2n}(v_0,v_1,v_2\ldots,v_{2n-1})\) by adding \(n-2\) chords between the vertex \(v_0\) and the vertices \(v_{2i-1}\) for \(1-i\delta n\). 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 \(G{2n_i,n_i,2,k,l}\), 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 \(C(2n_i,n_i-2)’s\), \(1- i\delta k\), 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 \(G{2n_i,n_i,2,k,l}\) as \(G{2n_i,n_i,2,k,l_c}\) if the path \(P_{2k-1}\) joins the vertices only at the common level \(l\) with chords.
In this paper, we show that \(G{2n_i,n_i,2,k,l_c}\) is graceful and admits an \(A\)-labeling, for \(k-\tau1, n_i\), \( 3,1\tau1,n_i\), and \(G{2n_i,n_i,2,k,1}\) is cordial, for \(n_i-n-3 ,k-1,1\tau i\).
1970-2025 CP (Manitoba, Canada) unless otherwise stated.