# Total-coloring of Sparse Graphs with Maximum Degree 6

Total-coloring of Sparse Graphs with Maximum Degree 6 Given a graph G = (V, E) and a positive integer k, a k-total-coloring of G is a mapping φ: V⋃E → {1, 2, ⋯, k} such that no two adjacent or incident elements receive the same color. The central problem of the total-colorings is the Total Coloring Conjecture, which asserts that every graph of maximum degree Δ admits a (Δ+2)-total-coloring. More precisely, this conjecture has been verified for Δ ≤ 5, and it is still open when Δ = 6, even for planar graphs. Let mad(G) denote the maximum average degree of the graph G. In this paper, we prove that every graph G with Δ(G) = 6 and mad(G) < 23/5 admits an 8-total-coloring. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Total-coloring of Sparse Graphs with Maximum Degree 6

, Volume 37 (4) – Oct 1, 2021
9 pages      /lp/springer-journals/total-coloring-of-sparse-graphs-with-maximum-degree-6-zmudLReqXx
Publisher
Springer Journals
Copyright © The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2021
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-021-1039-3
Publisher site
See Article on Publisher Site

### Abstract

Given a graph G = (V, E) and a positive integer k, a k-total-coloring of G is a mapping φ: V⋃E → {1, 2, ⋯, k} such that no two adjacent or incident elements receive the same color. The central problem of the total-colorings is the Total Coloring Conjecture, which asserts that every graph of maximum degree Δ admits a (Δ+2)-total-coloring. More precisely, this conjecture has been verified for Δ ≤ 5, and it is still open when Δ = 6, even for planar graphs. Let mad(G) denote the maximum average degree of the graph G. In this paper, we prove that every graph G with Δ(G) = 6 and mad(G) < 23/5 admits an 8-total-coloring.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Oct 1, 2021

Keywords: total-coloring; maximum average degree; discharging method; 05C15

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