New graph characteristic, the total H-irregularity strength of a graph, is introduced. Estimations on this parameter are obtained and for some families of graphs the precise values of this parameter are proved.
The study of valuations of graphs is a relatively young part of graph theory. In this article we survey what is known about certain graph valuations, that is, labeling methods: antimagic labelings, edge-magic total labelings and vertex-magic total labelings.
Open problem 1. For the paths Pₙ and the cycles Cₙ, determine if there is a vertex-antimagic total labeling for every feasible pair (a,d). Open problem 2. Apart from duality, how can a vertex-antimagic total labeling for a graph be used to construct another vertex-antimagic total labeling for the same graph, preferably with different a and d? Open problem 3. In Theorem 3, we found a way to construct VATL for a graph G from a vertex-magic total labeling of G. Are there other ways to do this? Open problem 4. Find, if possible, some structural characteristics of a graph which make a vertex-antimagic total labeling impossible
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