Signed Graphs: From Modulo Flows to Integer-Valued Flows
Jian Cheng, You Lu, Rong Luo, Cun‐Quan Zhang · SIAM Journal on Discrete Mathematics · 2018
Converting modulo flows into integer-valued flows is one of the most critical steps in the study of integer flows. Tutte and Jaeger's pioneering work shows the equivalence of modulo flows and integer-valued flows for ordinary graphs. However, such equivalence no longer holds for signed graphs. This motivates us to study how to convert modulo flows into integer-valued flows for signed graphs. In this paper, we generalize some early results by Xu and Zhang [ Discrete Math., 299 (2005), pp. 335--343], Schubert and Steffen [ European J. Combin., 48 (2015), pp. 34--47], and Zhu [ J. Combin. Theory Ser. B, 112 (2015), pp. 93--103] and show that, for signed graphs, every modulo $(2+\frac{1}{p})$-flow with $p \in {\mathbb Z}^+ \cup \{\infty\}$ can be converted/extended into an integer-valued flow.