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Choosability conjectures and multicircuits. (English) Zbl 0989.05041
For a graph $$G$$, let $$\chi(G)$$, $$\chi'(G)$$ and $$\chi''(G)$$ respectively be the ordinary vertex, edge and total chromatic number. Let $$\text{ch}(G)$$, $$\text{ch}'(G)$$ and $$ch''(G)$$ be the list (vertex), list (edge) and list total chromatic number of a graph $$G$$. Several old and new conjectures about choosability in graphs are discussed. Some of them are the following conjectures. The list-edge-colouring conjecture (LECC): For every multigraph $$G$$, $$\text{ch}'(G)=\chi'(G)$$. The list-total-colouring conjecture (LTCC): For every multigraph $$G$$, $$\text{ch}''(G)=\chi''(G)$$. The list-square-colouring conjecture (LSCC): For every graph $$G$$, $$\text{ch}(G^2)=\chi(G^2)$$, where $$G^2$$ is the square of $$G$$. In this paper LSCC and LTCC are verified for some classes of graphs which bring the authors to new conjectures.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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