Complex and Homomorphic Chromatic Number of Signed Planar Simple Graphs

We introduce the notion of complex chromatic number signed graphs as follows: given set $${\mathbb {C}}_{k,l}=\{\pm 1, \pm 2, \ldots , k\}\cup \{\pm 1i, 2i, li\}$$ where $$i=\sqrt{-1}$$ a graph $$(G,\sigma )$$ is said to be (k, l)-colorable if there exists mapping c vertices G {C}}_{k,l}$$ such that for every edge xy we have $$\begin{aligned} c(x)c(y)\ne \sigma (xy) |c(x)^2|. \end{aligned}$$ The $$(G, denoted $$\chi _{com}(G, defined smallest order admits l)-coloring. In this work, after providing an equivalent definition in language homomorphisms graphs, show are planar simple which not 4-colorable. That say: neither (2, 0)-colorable, nor (1, 1)-colorable, (0, 2)-colorable. 0)-colorable was subject conjecture by Máçajová, Raspaud and Škoviera recently disproved Kardoš Narboni using dual notion. provide direct approach short proof. 1)-colorable recent Jiang Zhu disprove work. Noting 2)-coloring same 0)-coloring -\sigma proves existence whose larger than 4. Further developing homomorphism approach, analogue 5-color theorem, find three minimal each on vertices, without $$K_1^{\pm }$$ (a vertex with both positive negative loop) having property admitting from graph. Finally identify several other problems high interest colorings graphs.