The (2,1)-Total Labeling Number of Outerplanar Graphs Is at Most Δ + 2

Abstract

A (2, 1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0, 1, .

Algorithmic aspects of distance constrained labeling: a survey

“…Also, Bazzaro et al [5] showed that λ T p,1 (G) ≤ ∆+p+s for any sdegenerated graph (by χ(G) ≤ s+1 and χ ′ (G) ≤ ∆+1 [62]), where an s-degenerated graph G is a graph which can be reduced to a trivial graph by successive removal of vertices with degree at most s, that λ T p,1 (G) ≤ ∆ + p + 3 for any planar graph (by the Four-Color Theorem), and that λ T p,1 (G) ≤ ∆ + p + 1 for any outerplanar graph other than an odd cycle (since any outerplanar graph is 2-degenerated, and any outerplanar graph other than an odd cycle satisfies χ ′ (G) = ∆ [28]). As for the (2, 1)-total labeling number of outerplanar graphs is known to be at most ∆ + 2, which is tight, i.e., there exists an outerplanar graph whose (2, 1)-total labeling number is ∆ + 2 [36,37]. Also, there are many related works about bounds on λ T p,1 (G) [17,38,52,56].…”