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

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.

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

“…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. Let ϕ be a function from E(G) to {1, 2, .…”

“…A critical (p, 1)-total k-labelled graph is a graph G such that it admits no (p, 1)-total k-labelling, and any proper subgraph of G has a (p, 1)-total k-labelling. Zhang, Yu and Liu [21] proved the following two structural theorems for the critical (p, 1)-total labelled graph. Remark: the original statements of Lemmas 6 and 7 in [10] are not as the same as the above two ones.…”

“…[21, Lemma 2.4] If G is a critical (p, 1)-total (M + 2p − 2)-labelled graph with maximum degree at most M, then there is no k-alternator F in G for any integer 2 ≤ k ≤ M+2p−22p If G is a 1-planar graph with ∆(G) ≥ 8p + 2 and p ≥ 2, then λ T p (G) ≤ ∆(G) + 2p − 2. Proof.…”

Light 3-Cycles in 1-Planar Graphs With Degree Restrictions

“…The notion of 1-planarity was introduced by Ringel [14], who proved that each 1-planar graph is vertex 7-colorable. This is the first result on the colorings of 1-planar graphs, and from then on, many authors started to investigate the coloring problems (see [1,3,4,6,15,16,17,21,23,24]) and the structural properties (see [5,8,9,10,11,13,18,19,20,22]) of 1-planar graphs. One of possible approaches in the study of local graph structures can be formalized in the following way (see [12]):…”

On the lightness of chordal 4-cycle in 1-planar graphs with high minimum degree

“…Zhang et al showed that each 1-planar graph G with maximum degree ∆ is ∆-edge-colorable if ∆ ≥ 10 [25], or ∆ ≥ 9 and G contains no chordal 5-cycles [19], or ∆ ≥ 8 and G contains no chordal 4-cycles [20], or ∆ ≥ 7 and G contains no 3-cycles [21], is (∆ + 1)-edge-choosable and (∆ + 2)-total-choosable if ∆ ≥ 16 [27], is ∆-edge-choosable and (∆ + 1)-total-choosable if ∆ ≥ 21 [27]. Zhang et al also showed that the (p, 1)-total labelling number of each 1-planar graph G is at most ∆(G) + 2p − 2 if ∆(G) ≥ 8p + 4 [28], and the linear arboricity of each 1-planar graph G is exactly ∆(G)/2 if ∆(G) ≥ 33 [24].…”