Total Irregularity Strength of Three Families of Graphs

Ramdani, Rismawati; Salman, A. N. M.; Assiyatun, Hilda; Semaničová-Feňovčíková, Andrea; Bača, Martin

doi:10.1007/s11786-015-0229-x

Cited by 12 publications

(7 citation statements)

References 16 publications

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“…Many other results e.g. for particular graph families can also be found in [4,22,26,29] and other papers.…”

Section: Introductionmentioning

confidence: 83%

Distant total irregularity strength of graphs via random vertex ordering

Przybyło

2018

Discrete Mathematics

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Let c : V ∪ E → {1, 2, . . . , k} be a (not necessarily proper) total colouring of a graph G = (V, E) with maximum degree ∆. Two vertices u, v ∈ V are sum distinguished if they differ with respect to sums of their incident colours, i.e. c(u) + e∋u c(e) = c(v) + e∋v c(e). The least integer k admitting such colouring c under which every u, v ∈ V at distance 1 ≤ d(u, v) ≤ r in G are sum distinguished is denoted by ts r (G). Such graph invariants link the concept of the total vertex irregularity strength of graphs with so called 1-2-Conjecture, whose concern is the case of r = 1. Within this paper we combine probabilistic approach with purely combinatorial one in order to prove that ts r (G) ≤ (2 + o(1))∆ r−1 for every integer r ≥ 2 and each graph G, thus improving the previously best result: ts r (G) ≤ 3∆ r−1 .Keywords: total vertex irregularity strength of a graph, 1-2 Conjecture, r-distant total irregularity strength of a graph

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“…Many other results e.g. for particular graph families can also be found in [4,22,26,29] and other papers.…”

Section: Introductionmentioning

confidence: 83%

Distant total irregularity strength of graphs via random vertex ordering

Przybyło

2018

Discrete Mathematics

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Complete bipartite graph is a totally irregular total graph

Tilukay

Taihuttu

Salman

et al. 2021

EJGTA

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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.

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“…Theorem 1.5 [12] Let be a graph of order . Let the sequence of Fibonacci numbers be defined by the recurrence relation = −1 + −2 , n ≥ 3, with seed values 1 = 1 and 2 = 2.…”

Section: Introductionmentioning

confidence: 99%

“…(8) In the same paper, [12], Ramdani et al determined the exact value of ( ) for G are gears, fungus, and copies of stars.…”

Section: Introductionmentioning

confidence: 99%

Total vertex irregularity strength of comb product of two cycles

Ramdani

Ramdhani

2018

MATEC Web Conf.

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Let G = (V (G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2...,k}. The vertex weight v under the labeling f is denoted by Wf(v) and defined by Wf(v) = f(v) + Σuv∈E(G)f(uv). A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labeling was introduced by Bača, Jendrol', Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H, in the vertex o, denoted by G⊳o H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of Cn and Cm where m ∈ {1,2}.

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Total Irregularity Strength of Three Families of Graphs

Cited by 12 publications

References 16 publications

Distant total irregularity strength of graphs via random vertex ordering

Distant total irregularity strength of graphs via random vertex ordering

Complete bipartite graph is a totally irregular total graph

Total vertex irregularity strength of comb product of two cycles

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