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One more probabilistic reformulation of the four colour conjecture. (English) Zbl 1261.05095
The author here gives another equivalent reformulation of the four colour conjecture (now a theorem!) for planar graphs: see for example his papers [Theory Probab. Appl. 48, No. 2, 368–372 (2003); translation from Teor. Veroyatn. Primen. 48, No. 2, 411–416 (2003; Zbl 1051.05044)] and [J. Graph Theory 46, No. 3, 167–179 (2004; Zbl 1053.05050)]. (In earlier work, he had given an equivalent formulation in terms of number theory: see e.g., R. Thomas [Notices Am. Math. Soc. 45, No. 7, 848–859 (1998; Zbl 0908.05040)]).
The reformulation here is in terms of the statement that, in every two-connected planar cubic graph \(G\) with \(3n\) edges, the event that two randomly selected orientations of the line graph are ‘congruent \(\mod 3\)’ - that is, they have the out-degrees in the two orientations congruent \(\mod 3\). Call this event \(B_{G}\). The result is that the four-color theorem is equivalent to the statement that \(B_{G}\) is not independent of the event \(A_{G}\), that two randomly selected orientations of the line graph have the same parity (in a suitable technical sense). More precisely, we get (with \(\chi_{G}(4)\) denoting the number of proper 4-colourings of \(G\)) \[ \mathbb{P}(B_{G}| A_{G})-\mathbb{P}(B_{G})=\frac{\chi_{G}(4)}{4}\left(\frac{27}{4096}\right)^{n}. \]
Some other reformulations are presented as well. The result seems not to be new: it appears to be equivalent to Theorem 5.4.4. in Andrew Goodall’s D.Phil thesis at Oxford in 2004 (unpublished, but see http://kam.mff.cuni.cz/~andrew/AJGthesis.pdf). However the proof here is non-trivially shorter.
MSC:
05C80 Random graphs (graph-theoretic aspects)
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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