The Total Irregularity Strength of Complete Graphs and Complete Bipartite Graphs

2020

IJC

Self Cite

<div class="page" title="Page 1"><div class="layoutArea"><div class="column">Under a totally irregular total k-labeling of a graph G = (V,E), we found that for some certain graphs, the edge-weight set W(E) and the vertex-weight set W(V) of G which are induced by k = ts(G), W(E) ∩ W(V) is a non empty set. For which k, a graph G has a totally irregular total labeling if W(E) ∩ W(V) = ∅? We introduce the total disjoint irregularity strength, denoted by ds(G), as the minimum value k where this condition satisfied. We provide the lower bound of ds(G) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.</div></div></div>

Section: Introductionmentioning

confidence: 96%

mentioning

confidence: 99%

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The total disjoint irregularity strength of some certain graphs

2020

IJC

Self Cite

“…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,…”

Section: Introductionmentioning

confidence: 99%

Complete bipartite graph is a totally irregular total graph

Taihuttu

et al. 2021

EJGTA

Self Cite

A graph G is called a totally irregular total k-graph if it has a totally irregular total k-labeling λ : V ∪ E → {1, 2, • • • , k}, that is a total labeling such that for any pair of different vertices x and y of G, their weights wt(x) and wt(y) are distinct, and for any pair of different edges e and f of G, their weights wt(e) and wt(f ) are distinct. The minimum value k under labeling λ is called the total irregularity strength of G, denoted by ts(G). For special cases of a complete bipartite graph K m,n , the ts(K 1,n ) and the ts(K n,n ) are already determined for any positive integer n. Completing the results, this paper deals with the total irregularity strength of complete bipartite graph K m,n for any positive integer m and n.

“…Combining previous conditions on irregular total labeling, Marzuki et al in [5] defined a totally irregular total labeling. Several upper bounds and exact values of the total irregularity strength of some classes were given in [5][6][7][8], and surveyed in [4].…”

Section: Introductionmentioning

confidence: 99%

The total face irregularity strength of some plane graphs

AKCE International Journal of Graphs and Combinatorics

Ilwaru

et al. 2020

A face irregular total k-labeling λ : V ∪ E → {1, 2, . . . , k} of a 2-connected plane graph G is a labeling of vertices and edges such that their face-weights are pairwise distinct. The weight of a face f under a labeling λ is the sum of the labels of all vertices and edges surrounding f . The minimum value k for which G has a face irregular total k-labeling is called the total face irregularity strength of G, denoted by t f s(G). The lower bound of t f s(G) is provided along with the exact value of two certain plane graphs. Improving the results, this paper deals with the total face irregularity strength of the disjoint union of multiple copies of a plane graph G. We estimate the bounds of t f s(G) and prove that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel.