Abstract The flow number of a signed graph ( G , Σ ) is the smallest positive integer k such that ( G , Σ ) admits a nowhere-zero integer k -flow. In 1983, Bouchet (JCTB) conjectured that every flow-admissible signed graph has flow number at most 6. This conjecture remains open for general signed graphs even for signed planar graphs. A Halin graph is a plane graph consisting of a tree without vertices of degree two and a circuit connecting all leaves of the tree. In this paper, we prove that every flow-admissible signed Halin graph has flow number at most 5, and determine the flow numbers of signed Halin graphs with a (3,1)-caterpillar tree as its characteristic tree.

中文翻译：

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带符号的 Halin 图的流数

摘要 有符号图的流数 ( G , Σ ) 是最小的正整数 k ，使得 ( G , Σ ) 允许无处为零的整数 k 流。1983 年，Bouchet (JCTB) 猜想每个流允许有符号图的流数最多为 6。这个猜想对于一般有符号图仍然是开放的，即使对于有符号平面图也是如此。Halin图是一个平面图，由一棵没有二阶顶点的树和一条连接树的所有叶子的电路组成。在本文中，我们证明了每个可允许流的有符号Halin图的流数最多为5，并以(3,1)-毛虫树为特征树确定有符号Halin图的流数。