an:05644620
Zbl 1221.05133
Cranston, Daniel W.
Multigraphs with \(\Delta \geq 3\) are totally-\((2\Delta - 1)\)-choosable
EN
Graphs Comb. 25, No. 1, 35-40 (2009).
00253131
2009
j
05C15
choosability; list coloring; total coloring; multigraph
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 \textit{O. V. Borodin}, \textit{A. V. Kostochka}, and \textit{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.]).
Zbl 0876.05032