Bermond, Jackson and Jaeger [J. Combin. Theory Ser. B 35 (1983): 297-308] proved that every bridgeless ordinary graph $G$ has a circuit $4$-cover and Fan [J. Combin. Theory Ser. B 54 (1992): 113-122] showed that $G$ has a circuit $6$-cover which together implies that $G$ has a circuit $k$-cover for every even integer $k\ge 4$. The only left case when $k = 2$ is the well-know circuit double cover conjecture. For signed circuit $k$-cover of signed graphs, it is known that for every integer $k\leq 5$, there are infinitely many coverable signed graphs without signed circuit $k$-cover and there are signed eulerian graphs that admit nowhere-zero $2$-flow but don't admit a signed circuit $1$-cover. Fan conjectured that every coverable signed graph has a signed circuit $6$-cover. This conjecture was verified only for signed eulerian graphs and for signed graphs whose bridgeless-blocks are eulerian. In this paper, we prove that this conjecture holds for signed $K_4$-minor-free graphs. The $6$-cover is best possible for signed $K_4$-minor-free graphs.
@article{10_37236_12572,
author = {You Lu and Rong Luo and Zhengke Miao and Cun-Quan Zhang},
title = {Signed circuit \(6\)-covers of signed {\(K_4\)-minor-free} graphs},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {2},
doi = {10.37236/12572},
zbl = {1564.05137},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12572/}
}
TY - JOUR
AU - You Lu
AU - Rong Luo
AU - Zhengke Miao
AU - Cun-Quan Zhang
TI - Signed circuit \(6\)-covers of signed \(K_4\)-minor-free graphs
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/12572/
DO - 10.37236/12572
ID - 10_37236_12572
ER -
%0 Journal Article
%A You Lu
%A Rong Luo
%A Zhengke Miao
%A Cun-Quan Zhang
%T Signed circuit \(6\)-covers of signed \(K_4\)-minor-free graphs
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/12572/
%R 10.37236/12572
%F 10_37236_12572
You Lu; Rong Luo; Zhengke Miao; Cun-Quan Zhang. Signed circuit \(6\)-covers of signed \(K_4\)-minor-free graphs. The electronic journal of combinatorics, Tome 32 (2025) no. 2. doi: 10.37236/12572
