On 3-total edge product cordial labeling of honeycomb

Peer review under responsibility of Kalasalingam University.

Figures & data

Fig. 4 The labeled segment $H_1$.

Fig. 5 The labeled segment $H_3$.

Fig. 6 The labeled segment $P_1$.

Fig. 7 The labeled segment $P_3$.

Fig. 8 The labeled segment $P_5$.

Fig. 9 The labeled segment $P_6$.

Fig. 10 The labeled segment $P_3^E$.

Fig. 14 The labeled segment $P_3^2$.

Fig. 15 The labeled segment $P_5^2$.

Fig. 1 The honeycomb $H_n^m$.

Fig. 2 The $3$-total edge product cordial labeling of $H_1^2\cong H_2^1$.

Fig. 3 The labeled segment $V_1$.

Fig. 11 The segment $V_n$ for $n$ odd.

Fig. 12 The segment $V_n$ for $n$ even.

Fig. 13 The labeled segment $V_2$.

Table 1 Numbers of 0s, 1s and 2s in the segments $P_1$, $P_3$, $P_5$, $P_6$ and $P_3^E$.

Segment	# of 0s	# of 1s	# of 2s
$P_1$	3	3	3
$P_3$	6	6	7
$P_5$	10	9	10
$P_6$	10	10	10
$P_3^E$	5	5	5

Table 2 Construction of the segment $V_n$.

$n$	Construction	# of 0s	# of 1s	# of 2s
$n=6t+1$, $t\ge 0$	$V_n=P_1{\oplus }_{v}t\cdot P_6$	$10t+3$	$10t+3$	$10t+3$
$n=6t+2$, $t\ge 1$	$V_n=P_5{\oplus }_v(t-1)\cdot P_6{\oplus }_v{P}_3^E$	$10t+5$	$10t+4$	$10t+5$
$n=6t+3$, $t\ge 0$	$V_n=P_3{\oplus }_{v}t\cdot P_6$	$10t+6$	$10t+6$	$10t+7$
$n=6t+4$, $t\ge 0$	$V_n=P_1{\oplus }_{v}t\cdot P_6{\oplus }_v{P}_3^E$	$10t+8$	$10t+8$	$10t+8$
$n=6t+5$, $t\ge 0$	$V_n=P_5{\oplus }_{v}t\cdot P_6$	$10t+10$	$10t+9$	$10t+10$
$n=6t+6$, $t\ge 0$	$V_n=P_3{\oplus }_{v}t\cdot P_6{\oplus }_v{P}_3^E$	$10t+11$	$10t+11$	$10t+12$

Fig. 16 The labeled segment $V_2^2$.

Table 3 Numbers of 0s, 1s and 2s in the segments $P_3^2$ and $P_5^2$.

Segment	# of 0s	# of 1s	# of 2s
$P_3^2$	12	13	13
$P_5^2$	20	19	19

Table 4 Construction of the segment $V_n^2$.

$n$	Construction	# of 0s	# of 1s	# of 2s
$n=6t+1$, $t\ge 0$	$V_n^2=\left[\begin{array}{c}\hfill V_n\hfill \\ \hfill {\oplus }_h\hfill \\ \hfill V_n\hfill \end{array}\right]$	$20t+6$	$20t+6$	$20t+6$
$n=6t+2$, $t\ge 1$	$V_n^2=P_5^2{\oplus }_v\left[\begin{array}{c}\hfill (t-1)\cdot P_6{\oplus }_v{P}_3^E\hfill \\ \hfill {\oplus }_h\hfill \\ \hfill (t-1)\cdot P_6{\oplus }_v{P}_3^E\hfill \end{array}\right]$	$20t+10$	$20t+9$	$20t+9$
$n=6t+3$, $t\ge 0$	$V_n^2=P_3^2{\oplus }_v\left[\begin{array}{c}\hfill t\cdot P_6\hfill \\ \hfill {\oplus }_h\hfill \\ \hfill t\cdot P_6\hfill \end{array}\right]$	$20t+12$	$20t+13$	$20t+13$
$n=6t+4$, $t\ge 0$	$V_n^2=\left[\begin{array}{c}\hfill V_n\hfill \\ \hfill {\oplus }_h\hfill \\ \hfill V_n\hfill \end{array}\right]$	$20t+16$	$20t+16$	$20t+16$
$n=6t+5$, $t\ge 0$	$V_n^2=P_5^2{\oplus }_v\left[\begin{array}{c}\hfill t\cdot P_6\hfill \\ \hfill {\oplus }_h\hfill \\ \hfill t\cdot P_6\hfill \end{array}\right]$	$20t+20$	$20t+19$	$20t+19$
$n=6t+6$, $t\ge 0$	$V_n^2=P_3^2{\oplus }_v\left[\begin{array}{c}\hfill t\cdot P_6{\oplus }_v{P}_3^E\hfill \\ \hfill {\oplus }_h\hfill \\ \hfill t\cdot P_6{\oplus }_v{P}_3^E\hfill \end{array}\right]$	$20t+22$	$20t+23$	$20t+23$