Entire Labeling of Plane Graphs

On face irregular evaluations of plane graphs

“…Immediately from Theorem 2 we get a lower bound for the entire face irregularity strength of a 2-connected plane graph. (2) fs This result was proved by Bača et al [8]. Furthermore, the existence of a face irregular entire 2-labeling of octahedron was shown in [9] and it proves the sharpness of the lower bound (2).…”

“…This bound was presented in [8]. Moreover, the exact value of the entire face irregularity strength for ladder L n P n P 2 , n ≥ 3, was determined in [8] and it proves the sharpness of the lower bound (3).…”

“…This bound was presented in [8]. Moreover, the exact value of the entire face irregularity strength for ladder L n P n P 2 , n ≥ 3, was determined in [8] and it proves the sharpness of the lower bound (3). Let us consider a simple graph G = (V, E) such that every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. We say that G admits an H-covering in this case.…”

“…The irregular labelings of plane graphs with restrictions placed on the weights of faces were introduced in [8] and there is defined the face irregularity strength of type (1, 1, 1) as entire face irregularity strength, denoted by efs(G). Some partial results on irregularity strength of type (1, 1, 0) under the notation total face irregularity strength can be found in [20].…”

“…The smallest face-weight under the face irregular k-labeling ϕ admits the value at least ϕ(x) +â(β + 1) + γ − 1. Since there are ∆(G) faces incident with x, it follows that the largest face-weight between them attains the value at least ϕ In [8] is proved that…”