Total Face Irregularity Strength of Certain Graphs

Algorithm 1

Total face irregularity strength of cycle-of-ladder.

Input: Cycle-of-ladder, , .

Algorithm: Begin labeling the 4-cycles in the ladders as follows.

Step 1: The labels of the bottom rung , the two parallel sides perpendicular to the bottom rung and the top rung constitute a 8-tuple divided as 3-tuple representing the labels of the bottom rung followed by 2-tuple representing the labels of the parallel edges followed by -tuple representing the labels of the top rung of the respective -cycle.

Step 2: We sequentially label the 4-cycles in each ladder, beginning from the bottom rung of , go up the ladder till all vertices and edges in are labeled, and repeat the same with in the same order.

Step 3: We list the labels of the first -cycles as

and the next eight -cycles as follows

Step 4: If the number of -cycles in is more than 9, continue labeling the subsequent cycles as , where is the -tuple obtained from by adding to each bit, .

Step 5: Repeat this procedure by adding to each , , till all the cycles are labeled.

Step 6: By our labeling, we get consecutive labels for each of the faces enclosed by these -cycles. Let denote the greatest of these face labels. Finally, we label the edge of the which are not yet labeled using the already used labels to arrive at a face label greater than .

Output:.

Proof of correctness: has number of -cycles. Then for every set of eight -cycles beginning from the second, considered sequentially from bottom to top and in the clockwise direction, the label is incremented by , beginning from . Hence the number of labels used is .