A note on complex-4-colorability of signed planar graphs

A pair (G,sigma) is called a signed graph if sigma : E(G) -> {1,-1} is a mapping which assigns to each edge e of G a sign sigma(e) is an element of{1,-1}. If (G,sigma) is a signed graph, then a complex-4-coloring of (G,sigma) is a mapping f : V(G) -> {1,-1,i,-i} with i = root-1 such that f(u)f(v) not equal sigma(e) for every edge e = uv of G. We prove that there are signed planar graphs that are not complex-4-colorable. This result completes investigations of Jin, Wong and Zhu as well as Jiang and Zhu on 4-colorings of generalized signed planar graphs disproving a conjecture of the latter authors.

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