(d,1)-total labelling of planar graphs with large girth and high maximum 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 such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least d. This notion was introduced in [F. Havet, (d, 1)-total labelling of graphs, in: Workshop on Graphs and Algorithms, Dijon, France, 2003]. In this paper, we prove that T d (G) + 2d − 2 for planar graphs with large girth and high maximum degree . Our results are optimal for d = 2.

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

“…Concerning the (p, 1)-total labelling, Bazzaro, Montassier and Raspaud [1] raised another interesting problem that is to answer when…”

“…Claim 16. Every (B, F, M )-face or (B, F, S)-face sends1 4 to its incident false vertex.Now we consider burdened 4 + -faces.Claim 17. Every burdened 4-face sends to each of its incident small vertices 3 4 if f is a special 4-face, 1 if f is an (F, S, B, S)-face, and at least 5 4 otherwise.…”

“…The structure around a false vertex v.(1.1) If at least two of vertices amongv 1 , v 2 , v 3 , v4 are big vertices, then we distinguish two subcases by symmetry. If v 1 and v 2 are big, then f 1 sends at least 1 to v by Claim 20(2), and each of f 2 and f 4 sends at least1 2 to v by Claim 20(1). This implies that c ′ (v) ≥ 4 − 6 + 1 and v 3 are big, then every face incident with v sends at least1 2 to v by Claim 20(1), which implies thatc ′ (v) ≥ 4 − 6 + 4 × If exactly one of the vertices among v 1 , v 2 , v 3 , v 4 , say v 1, is a big vertex, then v 2 and v 4 are middle vertices, because otherwise one vertex among v 2 and v 4 is a small vertex and the other is either middle or small, which is impossible since a small vertex is adjacent only to big vertices in G. By Claim 20(1), each of f 1 and f 4 sends at least 1 2 to v. If v 3 is a small vertex, then each of f 2 and f 3 sends1 2 to v by Claim 20(3), which implies thatc ′ (v) ≥ that v3 is a middle vertex.…”

“…If v 1 and v 2 are big, then f 1 sends at least 1 to v by Claim 20(2), and each of f 2 and f 4 sends at least1 2 to v by Claim 20(1). This implies that c ′ (v) ≥ 4 − 6 + 1 and v 3 are big, then every face incident with v sends at least1 2 to v by Claim 20(1), which implies thatc ′ (v) ≥ 4 − 6 + 4 × If exactly one of the vertices among v 1 , v 2 , v 3 , v 4 , say v 1, is a big vertex, then v 2 and v 4 are middle vertices, because otherwise one vertex among v 2 and v 4 is a small vertex and the other is either middle or small, which is impossible since a small vertex is adjacent only to big vertices in G. By Claim 20(1), each of f 1 and f 4 sends at least 1 2 to v. If v 3 is a small vertex, then each of f 2 and f 3 sends1 2 to v by Claim 20(3), which implies thatc ′ (v) ≥ that v3 is a middle vertex. If f 2 and f 3 are 3-faces so that v 3 is an M 8− -vertex, v 2 and v 4 are M 9+ -vertices, then each of f 1 and f 4 sends at least to v by Claim 20(4), and f 2 and f 3 totally sends 2 3 to v by Claim 20(5), which implies that c ′ (v) ≥ 4 − 6 + 2 × Claim 20(5), f 2 and f 3 totally sends at least 1 to v, and thus c ′ (v) ≥ If none of the vertices amongv 1 , v 2 , v 3 , v4 is a big vertex, then they are all middle vertices.…”

(2,1)-Total labeling of cycle with parallel paths

“…Also, (p, 1)-total number is determined for classes of special graphs. For example, the (p, 1)-total number is determined for complete graphs [6], planar graphs [1], graphs with a given maximum average degree [8], outer planar graphs [2,11], etc. The case p = 1 corresponds to the usual notion of total colouring, which is NP-hard to compute even for cubic bipartite graphs.…”