|
| 摘要: |
| 本文研究了符号图的符号线图的无零流问题. 利用数学归纳法以及通过对符号图结构的分析, 获得了当符号图的底图为简单图且没有~$2$~度点时, 其符号线图允许一个无零~$4$~流的结果. 特别地, 如果符号图的底图为简单图且没有~$2$~度点和~$4$~度点, 那么它的符号线图允许一个无零~$3$~流. 验证了对具有上述结构的符号图的符号线图, Bouchet~$6$~猜想是成立的. 同时对于连通的~$2$~正则符号圈~$(C_{n},\sigma)$, 根据~$n$~的奇偶性, 我们研究了~$(C_{n},\sigma)$~的符号线图是否是流允许的. |
| 关键词: 符号图 符号线图 无零流 |
| DOI： |
| 分类号:O157.5 |
| 基金项目:国家自然科学基金资助 (12061060, 11971196, 11961007) |
|
| NOWHERE-ZERO FLOWS ON THE SIGNED LINE GRAPHS OF SIGNED GRAPHS |
|
HE Jing,ZHANG Chao
|
| Abstract: |
| This paper studies the nowhere-zero flow of the signed line graphs of signed graphs. By using mathematical induction and analyzing the structure of signed graphs, we prove that when the underlying graphs of the signed graphs are simple and have no vertices of degree $2$, the signed line graphs admit a nowhere-zero $4$-flow. In particular, if the underlying graphs of the signed graphs are simple without vertices of degree $2$ and $4$, then the signed line graphs admit a nowhere-zero $3$-flow. It is verified that for signed graphs with the aforementioned structures, their signed line graphs satisfy Bouchet"s $6$-flow conjecture.
For a connected $2$-regular signed cycle $(C_{n},\sigma)$, based on the parity of $n$, we investigate whether the signed line graph of $(C_{n},\sigma)$ is flow-admissible. |
| Key words: signed graphs signed line graph nowhere-zero flow |
