Geodesic
Total Coloring of Claw-Free Planar Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 771-777

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A total coloring of a graph is an assignment of colors to both its vertices and edges so that adjacent or incident elements acquire distinct colors. Let Δ(G) be the maximum degree of G. Vizing conjectured that every graph has a total (Δ + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs, for which the only open case is Δ = 6. Claw-free planar graphs have Δ ≤ 6. In this paper, we prove that the Total Coloring Conjecture holds for claw-free planar graphs.
Keywords: total coloring, total coloring conjecture, planar graph, claw 
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Liang, Zuosong. Total Coloring of Claw-Free Planar Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 771-777. http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a5/