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An upper bound for the total chromatic number. (English) Zbl 0725.05043
The chromatic number, the edge chromatic number, and the total chromatic number of a graph H are respectively denoted by $$\chi$$ (H), $$\chi '(H)$$, and $$\chi ''(H)$$. Theorem: For any graph H, $$\chi ''(H)\leq \chi '(H)+2\lceil \sqrt{\chi (H)}\rceil.$$ Let $$\Delta$$ (H) be the maximum degree of graph H. The proof of the theorem relies on the following interesting lemma: Let $$H=(V,E)$$ be a graph with $$\chi '(H)=\Delta (H)$$ and W be an independent subset of V. Any coloring of W with $$\Delta$$ (H) colors can be extended to a proper coloring of $$W\cup E$$ with $$\Delta (H)+1$$ colors such that if edge e is colored with the new color, then one of the vertices incident with e is in W.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
edge chromatic number; total chromatic number
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##### References:
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