Extending Grötzsch's 3-coloring theorem in the flow setting, Steinberg and Younger in 1989 proved that every 4-edge-connected planar or projective planar graph admits a nowhere-zero 3-flow (3-NZF for short), while Tutte's 3-flow conjecture asserts all 4-edge-connected graphs admit 3-NZFs. In this paper, we generalize Grötzsch's theorem to signed planar graphs by showing that every 4-edge-connected signed planar graph with two negative edges admits a 3-NZF. On the other hand, a result from Máčajová and Škoviera implies that there exist infinitely many 4-edge-connected signed planar graphs with three negative edges admitting no 3-NZFs but permitting 4-NZFs. Our proof employs the flow extension ideas from Steinberg-Younger and Thomassen, as well as refined exploration of the location of negative edges and elaborated discharging arguments in signed planar graphs.
@article{10_37236_11892,
author = {Jiaao Li and Yulai Ma and Zhengke Miao and Yongtang Shi and Weifan Wang},
title = {Nowhere-zero 3-flows in signed planar graphs},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {2},
doi = {10.37236/11892},
zbl = {1543.05036},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11892/}
}
TY - JOUR
AU - Jiaao Li
AU - Yulai Ma
AU - Zhengke Miao
AU - Yongtang Shi
AU - Weifan Wang
TI - Nowhere-zero 3-flows in signed planar graphs
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11892/
DO - 10.37236/11892
ID - 10_37236_11892
ER -
%0 Journal Article
%A Jiaao Li
%A Yulai Ma
%A Zhengke Miao
%A Yongtang Shi
%A Weifan Wang
%T Nowhere-zero 3-flows in signed planar graphs
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11892/
%R 10.37236/11892
%F 10_37236_11892
Jiaao Li; Yulai Ma; Zhengke Miao; Yongtang Shi; Weifan Wang. Nowhere-zero 3-flows in signed planar graphs. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/11892
