-labeling of supersubdivided connected graph plus an edge

Figures & data

Fig. 1 (a) The graph $G$ (b) Supersubdivision graph of the graph $G$.

Table 1 Value of $m_1$ depending on the congruence class of $n+2$ modulo 4.

Congruence class of $n+2$ mod 4	$m_1$
$4k$, $k\ge 2$	$2k-1$
$4k+1$, $k\ge 2$	$2k$
$4k+2$, $k\ge 2$	$2k+1$
$4k+3$, $k\ge 2$	$2k+2$

Table 2 Value of $m_2$ depending on the nature of $n$.

Nature of $n$	$m_2$
$6\le n\le 9$	2
$n\ge 10$	$\lfloor \frac{n+2}{4}\rfloor$

Fig. 2 Labels of vertices and edges of $K_{2,m_1}$ and $K_{2,m_2}$.

Table 3 Value of $a$ depending on the congruence class of $n+2$ modulo 4.

Congruence class $n+2$ mod 4	$a$
$4k$, $k\ge 2$	$2k-2$
$4k+1$, $k\ge 2$	$2k-1$
$4k+2$, $k\ge 2$	$2k$
$4k+3$, $k\ge 2$	$2k+1$

Fig. 3 Labels of vertices and edges of $K_{2,t_{ij}{r}_{ij}}$ for each $i$, $1\le i\le n-2$ and for $j$, $1\le i\le p_i$.

Fig. 4 Labels of vertices and edges of the complete bipartite graph which replaces the edge $v_0{v}_1$.

Fig. 5 The connected graph $G$ with edge labels.

Fig. 6 $\gamma$-labeling of $SSD\left(G\right)+\hat{e}$.