(d,1)‐total labeling of graphs with a given maximum average degree

Abstract

The (d,1)-total number T d (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same color, no two incident edges have the same color, and the distance between the color of a vertex and its incident edges is at least d. In this paper, we prove that

(d,1)-total labelling of planar graphs with large girth and high maximum degree

“…In a companion paper, we will give the relationship between the (d, 1)-total labelling of a graph G and its maximum average degree [12].…”

Algorithmic aspects of distance constrained labeling: a survey

“…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]. From the algorithmic point of view, Havet and Thomassé [41] showed that for bipartite graphs, if (i) p ≥ ∆ or (ii) ∆ = 3 and p = 2, then the (p, 1)-total labeling problem is polynomially solvable and otherwise it is NP-hard.…”

On (p, 1)-total labelling of planar graphs

“…In Yu et al (2011), Yu et al proved that every planar graph with maximum degree ≥ 12 satisfies that + 1 ≤ λ T 2 ≤ + 2. The ( p, 1)-total labelling conjecture in general has been considered for some other classes such as planar graphs with high girth and high maximum degree (Bazzaro et al 2007) and graphs with a given maximum average degree (Montassier and Raspaud 2006). Particularly, Bazzaro, Montassier and Raspaud proved the following theorem for all planar graphs (Bazzaro et al 2007).…”