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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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