On The Total Irregularity Strength of Regular Graphs

Abstract

Abstract. Let = ( , ) be a graph. A total labeling : ∪ → {1, 2, ⋯ , } is called a totally irregular total -labeling of if every two distinct vertices and in satisfy ( ) ≠ ( ) and every two distinct edges 1 2 and 1 2 in satisfyThe minimum for which a graph has a totally irregular total -labeling is called the total irregularity strength of , denoted by ( ). In this paper, we consider an upper bound on the total irregularity strength of copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total -labeling of a regular graph and we consider the total irregularity strength of copies of a path on two vertices, copies of a cycle, and copies of a prism 2 .

Complete bipartite graph is a totally irregular total graph

“…By Equations ( 7), ( 8), (9), and Theorem 2.1, we can conclude that complete bipartite graph K m,n for any positive integer m and n is a totally irregular total graph with , otherwise.…”

“…For several cartesian product graphs, Ramdani and Salman in [8], showed that the lower bound (6) is sharp. Later, Ramdani, Salman, and Assiyatun [9] gave an upper bound of ts for some regular graphs. Then, in [10], Ramdani, Salman, Assiyatun, Semaničová-Feňovčíková, and Bača, proved that gear graphs, fungus graphs, ts(F g n ), for n even, n ≥ 6; and disjoint union of stars are totally irregular total graphs with the ts equal to their tes.…”

“…Next, Tilukay, Tomasouw, Rumlawang, and Salman in [13] found that complete graph K n and complete bipartite graph K n,n are both totally irregular total graphs with their ts equal to the tes. They [9] obtained that for any positive integer n ≥ 2,…”

Total vertex irregularity strength of comb product of two cycles

“…In [15], Ramdani et al determined the total irregularity strength of regular graphs. In the paper, they gave an upper bound on total irregularity strength of copies of arbitrary graph .…”

On H-irregularity strengths of G-amalgamation of graphs

“…In [21] there are given upper and lower bounds for the parameter ts(G). Ramdani and Salman in [24] determined the exact values of the total irregularity strength for several Cartesian product graphs.…”