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Multigraphs with $$\Delta \geq 3$$ are totally-$$(2\Delta - 1)$$-choosable. (English) Zbl 1221.05133
Summary: The total graph $$T(G)$$ of a multigraph $$G$$ has as its vertices the set of edges and vertices of $$G$$ and has an edge between two vertices if their corresponding elements are either adjacent or incident in $$G$$. We show that if $$G$$ has maximum degree $$\Delta (G)$$, then $$T(G)$$ is $$(2\Delta (G) - 1)$$-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for $$\Delta (G) > 3$$ was $$\lfloor{\frac{3}{2}\Delta(G)+2 \rfloor}$$, by O. V. Borodin, A. V. Kostochka, and D. R. Woodall [“List edge and list total colourings of multigraphs,” J. Comb. Theory, Ser. B 71, No.2, 184–204 (1997; Zbl 0876.05032)]. When $$\Delta (G) = 4$$, our algorithm gives a better upper bound. When $$\Delta (G)\in \{3,5,6\}$$, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of [loc. cit.]).
##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
choosability; list coloring; total coloring; multigraph
Full Text:
##### References:
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