Abstract
For a signed graph $G$ and non-negative integer $d$, it was shown by DeVos et al. that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $\Gamma$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $\Gamma$ of order $k$ with $\epsilon(\Gamma)=d$, where $\epsilon(\Gamma)$ is the largest integer $d$ for which $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We define a class of particular directed circuits in $G$, namely the fundamental directed circuits, and show that all $\Gamma$-flows (not necessarily nowhere-zero) in $G$ can be generated by these circuits. It turns out that all $\Gamma$-flows in $G$ can be evenly partitioned into $2^{\epsilon(\Gamma)}$ classes specified by the elements of order 2 in $\Gamma$, each class of which consists of the same number of flows depending only on the order of $\Gamma$. Using an extension of Whitney's broken circuit theorem of Dohmen and Trinks, we give a combinatorial interpretation of the coefficients in $F_d(G,x)$ for $d=0$ in terms of broken bonds. Finally, we show that the sets of edges in a signed graph that contain no broken bond form a homogeneous simplicial complex.
Highlights
Nowhere-zero Zk-flows, or modular k-flows, in a graph were initially introduced by Tutte [17] as a dual problem to vertex-colouring of plane graphs
It has long been known that the number of nowhere-zero Zk-flows, or, more generally, nowhere-zero Γ-flows for an Abelian group Γ of order k is a polynomial function in k, which does not depend on the algebraic structure of the group [17]
In a similar way to flows in plane graphs, or more generally in graphs embedded in an orientable surface, the definition of Zk-flows in signed graphs is naturally considered for the study of graphs embedded in a non-orientable surface, where nowherezero Zk-flows emerge as the dual notion to local tensions [2, 13]
Summary
Nowhere-zero Zk-flows, or modular k-flows, in a graph were initially introduced by Tutte [17] as a dual problem to vertex-colouring of plane graphs. Each class consists of the same number of flows, which depends only on the order of the group This gives an explanation for why the number of the Γ-flows in a signed graph varies with different (Γ) and, gives an answer to a problem posed by Beck and Zaslavsky in [1]. This result yields an explicit expression of the polynomial Fd(G, x) obtained earlier by Goodall et al. In the fifth section we give a combinatorial interpretation of the coefficients in Fd(G, x) for d = 0. The coefficients of F0(G, x) are the simplex counts in each dimension of the complex
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.