A pair $(G,\sigma)$ is called a {\it signed graph} if $\sigma: E(G) \longrightarrow \{1,-1\}$ is a mapping which assigns to each edge $e$ of $G$ a sign $\sigma(e) \in \{1,-1\}$. If $(G,\sigma)$ is a signed graph, then a {\it complex-4-coloring} of $(G,\sigma)$ is a mapping $f: V(G) \longrightarrow \{1,-1,i,-i\}$ with $i=\sqrt{-1}$ such that $f(u)f(v) \not= \sigma(e)$ 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.
@article{10_37236_9844,
author = {Arnfried Kemnitz and Margit Voigt},
title = {A note on complex-4-colorability of signed planar graphs},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {2},
doi = {10.37236/9844},
zbl = {1464.05154},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9844/}
}
TY - JOUR
AU - Arnfried Kemnitz
AU - Margit Voigt
TI - A note on complex-4-colorability of signed planar graphs
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/9844/
DO - 10.37236/9844
ID - 10_37236_9844
ER -
%0 Journal Article
%A Arnfried Kemnitz
%A Margit Voigt
%T A note on complex-4-colorability of signed planar graphs
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/9844/
%R 10.37236/9844
%F 10_37236_9844
Arnfried Kemnitz; Margit Voigt. A note on complex-4-colorability of signed planar graphs. The electronic journal of combinatorics, Tome 28 (2021) no. 2. doi: 10.37236/9844
