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The total chromatic number of graphs of high minimum degree. (English) Zbl 0753.05033
Summary: If $$G$$ is a simple graph with minimum degree $$\delta(G)$$ satisfying $$\delta(G)\geq{5\over 6}(| V(G)|+1)$$ the total chromatic number conjecture holds; moreover if $$\delta(G)\geq{3\over 4}| V(G)|$$ then $$\chi_ T(G)\leq\Delta(G)+3$$. Also if $$G$$ has odd order and is regular with $$d(G)\geq{1\over 3}\sqrt 7| V(G)|$$ then a necessary and sufficient condition for $$\chi_ T(G)=\Delta(G)+1$$ is given.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
total chromatic number; graph of high minimum degree
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