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

Abstract

1 if e 1 and e 2 are two adjacent edges in G and | f (u) − f (e)| ≥ p if the vertex u is incident with the edge e. The minimum k such that

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

“…As we know, λ T p (G) ≤ ∆(G) + 2p − 2 for planar graphs with maximum degree ∆(G) ≥ 4p + 4 and p ≥ 2, which is a result due to Sun and Wu [14]. In this paper, we prove λ T p (G) ≤ ∆(G) + 2p − 2 for 1-planar graphs with maximum degree ∆(G) ≥ 6p + 7 and p ≥ 2.…”

“…Yu et al [17] proved for planar graphs with maximum degree ∆(G) ≥ 12 that λ T 2 (G) ≤ ∆(G) + 2. Recently, Sun and Wu [14] proved that λ T p (G) ≤ ∆(G) + 2p − 2 for planar graphs with maximum degree ∆(G) ≥ 4p + 4 and p ≥ 2, which improved both the result of Bazzaro, Montassier and Raspaud, and the result of Yu et al mentioned above. The (p, 1)-total labelling of 1-planar graphs was first considered in 2011 by Zhang, Yu and Liu [21].…”

A Structure of 1-Planar Graph and Its Applications to Coloring Problems

“…Concerning this problem, Bazzaro, Montassier and Raspaud [1] proved that if G is a planar graph with ∆(G) ≥ 8p + 2 and p ≥ 2, then λ T p (G) ≤ ∆(G) + 2p − 2. The lower bound for the maximum degree in this result was recently improved to 4p + 4 by Sun and Wu [14]. For 1-planar graphs, Zhang, Yu and Liu [21] proved the following result.…”

“…The lower bound for the maximum degree in this result was recently improved to 4p + 4 by Sun and Wu [14]. For 1-planar graphs, Zhang, Yu and Liu [21] proved the following result.…”

Número (p,1)-total de near-ladders e Petersen generalizados

“…Para p arbitrário, λ t p (G) está determinado para algumas classes de grafos, dentre elas: ciclos (Havet e Yu, 2008), Flower Snarks (Chunling et al, 2010 e grafos Sierpiński-Like (Deng et al, 2019). Boa parte dos resultados conhecidos abordam limitantes para λ t p (G) restritos a classes de grafos (Montassier e Raspaud, 2006;Sun e Wu, 2017). No caso dos regulares, sabe-se que λ t p (G) ≥ ∆ + p, p ≥ 2 (Havet e Yu, 2008).…”