Counterexamples to the total vertex irregularity strength’s conjectures

The total vertex irregularity strength tvs( G ) of a simple graph G ( V, E ) is the smallest positive integer k so that there exists a function ϕ : V ∪ E → [1 , k ] provided that all vertex-weights are distinct, where a vertex-weight is the sum of labels of a vertex and all of its incident edges. In the paper [Nurdin, E. T. Baskoro, A. N. M. Salman, N. N. Gaos, Discrete Math. 310 (2010) 3043–3048], two conjectures regarding the total vertex irregularity strength of trees and general graphs were posed as follows: (i) for every tree T , tvs( T ) = max {(cid:100) ( n 1 + 1) / 2 (cid:101) , (cid:100) ( n 1 + n 2 + 1) / 3 (cid:101) , (cid:100) ( n 1 + n 2 + n 3 + 1) / 4 (cid:101)} , and (ii) for every graph G with minimum degree δ and maximum degree ∆ , tvs( G ) = max {(cid:100) ( δ + (cid:80) i j =1 n j ) / ( i + 1) (cid:101) : i ∈ [ δ, ∆] } , where n j denotes the number of vertices of degree j . In this paper, we disprove both of these conjectures by giving infinite families of counterexamples.