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Colouring of generalized signed triangle-free planar graphs. (English) Zbl 1403.05049
Summary: We view an undirected graph $$G$$ as a symmetric digraph, where each edge $$x y$$ is replaced by two opposite arcs $$e = (x, y)$$ and $$e^{- 1} = (y, x)$$. Assume $$S$$ is an inverse closed subset of permutations of positive integers. We say $$G$$ is $$S$$-$$k$$-colourable if for any mapping $$\sigma : E(G) \rightarrow S$$ with $$\sigma(x, y) = (\sigma(y, x))^{- 1}$$, there is a mapping $$f : V(G) \rightarrow [k] = \{1, 2, \dots, k \}$$ such that $$\sigma_e(f(x)) \neq f(y)$$ for each arc $$e = (x, y)$$. The concept of $$S$$-$$k$$-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets $$S$$ such that every triangle-free planar graph is $$S$$-3-colourable. Such a set $$S$$ is called TFP-good. Grötzsch’s theorem is equivalent to say that $$S = \{\mathrm{id} \}$$ is TFP-good. We prove that for any inverse closed subset $$S$$ of $$S_3$$ which is not isomorphic to $$\{\mathrm{id},(12) \}$$, $$S$$ is TFP-good if and only if either $$S = \{\mathrm{id} \}$$ or there exists $$a \in [3]$$ such that for each $$\pi \in S$$, $$\pi(a) \neq a$$. It remains an open question to determine whether or not $$S = \{\mathrm{id},(12) \}$$ is TFP-good.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C22 Signed and weighted graphs 05C20 Directed graphs (digraphs), tournaments
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