A note on complex-4-colorability of signed planar graphs (Q831346)

A pair \((G, \sigma )\) is called a signed graph if \(\sigma: E(G) \to\{1,-1\}\) is a mapping which assigns to each edge \(e\) of \(G\) a sign \(\sigma(e)\in\{1,-1\}\). \textit{T. Zaslavsky} [Discrete Math. 39, 215--228 (1982; Zbl 0487.05027)] introduced coloring of signed graph, after many works are undertaken by researchers and various papers published. If \((G,\sigma)\) is a signed graph, then a complex-4-coloring of \((G,\sigma)\) is a mapping \( f:V(G)\to\{1,-1,i,-i\}\) with \(i =\sqrt{-1}\) such that \(f(u)f(v) \not=\sigma(e)\) for every edge \(e=uv\) of \(G\). Here, the authors prove that there are signed planar graphs that are not complex-4-colorable. This result completes investigations of \textit{L. Jin} et al. [Eur. J. Comb. 92, Article ID 103198, 6 p. (2021; Zbl 1458.05072)] as well as \textit{Y. Jiang} and \textit{X. Zhu} [Electron. J. Comb. 27, No. 3, P3.31, 23 p. (2020; Zbl 1446.05032)] on 4-colorings of generalized signed planar graphs disproving a conjecture of \textit{L. Jin} et al. [``Colouring of generalized signed planar graphs'', Preprint, \url{arXiv:1811.08584}]. This paper contains a useful information to researchers working in the areas of coloring and signed graph topics.