This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each k ∈ {3, 4, 5, 6}, every signed planar graph without circuits of length k is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let E i be the set of edges contained in precisely i members of the k 1-factors. Let µ k (G) be the smallest |E 0 | over all lists of k 1-factors of G. We study lists by three 1-factors, and call

Assume G is a graph and S is a set of permutations of integers. An S-labeling of G is a pair (D, σ), where D is an orientation of G and σ ∶ E(D) → S is a mapping which assigns to each arc e = (u, v) of D a permutation σ e ∈ S.The concept of S-kcolouring is a common generalization of many colouring concepts, including kcolouring, signed k-colouring, signed Z k -colouring, DP-k-colouring, group colouring and colouring of gained graphs. We are interested in the problem as for which subset S of S 4 , every planar graph is S-4-colourable. We call such a subset S a good subset. The famous Four Colour Theorem is equivalent to say that S = {id} is good. A result of Král, Pangrác and Voss is equivalent to say that S = {id, (1234), (13)(24), (1432)} and S = {id, (12)(34), (13)(24), (14)(23)} are not good. These results are strengthened by a very recent result of Narboni and Tarkos, which implies that S = {id, (12)(34)} is not good and another very recent result of Zhu which implies that S = {id, (12)} is not good. This paper proves if S is a susbet of S 4 containing id, then S is good if and only if S = {id}.

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