Matt DeVos, Jiaao Li, You Lu, Rong Luo, Cun-Quan Zhang, Zhang Zhang
In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero $6$-flow. Bouchet himself proved that such signed graphs admit nowhere-zero $216$-flows and Zyka further proved that such signed graphs admit nowhere-zero $30$-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero 11-flow.
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Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic signed graph admits a nowhere-zero $10$-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $10$-flow. As a byproduct, we also show that every flow-admissible hamiltonian signed graph admits a nowhere-zero $8$-flow.
Nowhere-zero 8-flows in cyclically 5-edge-connected, flow-admissible signed graphs
本文证明在循环5边连通的双向图中，Bouchet关于无处为零流的猜想中的216可改进为8。
An 8-flow theorem for signed graphs
本文证明了流容许且基础图容许无处为零4-流的符号图存在无处为零8-流，结合四色定理推广了Li等人关于符号3-边可着色三次图和符号哈密顿图的先前结论。
Nowhere-zero 8-flows in 3-edge-connected signed graphs
本文证明了3-edge-connected的流可接受签名图存在无零8流，改进了Bouchet关于签名图无零流的猜想。
Bouchet's conjecture for cyclically 5-edge-connected, cubic signed graphs
该研究证明了布切特猜想在循环五边连通立方符号图中成立，表明此类图具有处处非零的六流。
Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.
Nowhere-zero flows on signed supereulerian graphs
本文验证了Bouchet关于带符号图无处为零6-流的猜想在具有生成偶欧拉子图的图类上成立，并通过构造保号双射证明其紧性，且推广至阿贝尔凯莱图。
A signed graph is a graph $G$ associated with a mapping $σ: E(G)\to \{-1,+1\}$, denoted by $(G,σ)$. A $cycle$ of $(G,σ)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges, and negative otherwise. A $circuit$ of of a signed graph $(G,σ)$ is a positive cycle or a barbell consisting of two edge-disjoint negative cycles joined by a path. The definition of a circuit of signed graph comes from the signed-graphic matroid. A circuit cover of $(G,σ)$ is a family of circuits covering all edges of $(G,σ)$. A circuit cover with the smallest total lengt…
A signed circuit is a minimal signed graph (with respect to inclusion) that admits a nowhere-zero flow. We show that each flow-admissible signed graph on $m$ edges can be covered by signed circuits of total length at most $(3+2/3)\cdot m$, improving a recent result of Cheng et al. [manuscript, 2015]. To obtain this improvement we prove several results on signed circuit covers of trees of Eulerian graphs, which are connected signed graphs such that removing all bridges results in a collection of Eulerian graphs.
Many basic properties in Tutte's flow theory for unsigned graphs do not have their counterparts for signed graphs. However, signed graphs without long barbells in many ways behave like unsigned graphs from the point view of flows. In this paper, we study whether some basic properties in Tutte's flow theory remain valid for this family of signed graphs. Specifically let $(G,σ)$ be a flow-admissible signed graph without long barbells. We show that it admits a nowhere-zero $6$-flow and that it admits a nowhere-zero modulo $k$-flow if and only if it admits a nowhere-zero integer $k$-flow for each …
Nowhere-zero 5-flow on signed ladders
本文针对Bouchet提出的流可行有符号图存在无处为零6-流的猜想，证明有符号梯子图、环形梯子图与Möbius梯子图均满足该猜想，且除一类有符号环形梯子图外，其余均存在无处为零5-流。
Group connectivity of 3-edge-connected signed graphs
本文研究符号图的 $A$-连通性，通过构造 Jaeger 等人三次图分解的符号图版本，证明3-边连通且允许流的符号图在 $|A|\geq 6$ 且 $|A|\neq 7$ 时为 $A$-连通。
A {sign-circuit cover} $\mathcal{F}$ of a signed graph $(G, σ)$ is a family of sign-circuits which covers all edges of $(G, σ)$. The shortest sign-circuit cover problem was initiated by Má$\check{\text{c}}$ajová, Raspaud, Rollová, and Škoviera (JGT 2016) and received many attentions in recent years. In this paper, we show that every flow-admissible 3-edge colorable cubic signed graph $(G, σ)$ has a sign-circuit cover with length at most $\frac{20}{9} |E(G)|$.
We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained parameter we provide some examples of signed graphs having two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of each mentioned valency with only two distinct eigenvalues. We prove that for any $k\geq 5$ there are infinitely many connected signed $k$-regular graphs having maximum eigenvalue $\sqrt{k}$. Moreover for each $m\ge…
This paper is devoted to a detailed study of nowhere-zero flows on signed eulerian graphs. We generalise the well-known fact about the existence of nowhere-zero $2$-flows in eulerian graphs by proving that every signed eulerian graph that admits an integer nowhere-zero flow has a nowhere-zero $4$-flow. We also characterise signed eulerian graphs with flow number $2$, $3$, and $4$, as well as those that do not have an integer nowhere-zero flow. Finally, we discuss the existence of nowhere-zero $A$-flows on signed eulerian graphs for an arbitrary abelian group~$A$.
We survey known results related to nowhere-zero flows and related topics, such as circuit covers and the structure of circuits of signed graphs. We include an overview of several different definitions of signed graph colouring.
This paper proves that for any positive integer $k$, every essentially $(2k+1)$-unbalanced $(12k-1)$-edge connected signed graph has circular flow number at most $2+\frac 1k$.
Bouchet conjectured in 1983 that each signed graph that admits a nowhere-zero flow has a nowhere-zero 6-flow. We prove that the conjecture is true for all signed series-parallel graphs. Unlike the unsigned case, the restriction to series-parallel graphs is nontrivial; in fact, the result is tight for infinitely many graphs.
We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13--28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which occur in two types -- either balanced circuits or pairs of disjoint unbalanced circuits connected with a path intersecting them only at its ends. As an application of this result we show that a signed graph $G$ admitting a nowhere-zero $k$-flow has a covering with signed circuits of total length at most $2(k-…
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 does not hold any more 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), Schubert and Steffen (European J. Combin.~48, 2015), and Zhu (J. Combin. Theory Ser. B~112, 2015), and sho…
It is well known that each nonnegative integral flow on a graph can be decomposed into a sum of nonnegative graphic circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that the indecomposable flows on graphs are those graphic circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than those of unsigned graphs. This paper gives a complete description of indecomposable flows on signed graphs from the viewpoint of resolution of singularities by means of double covering graphs.
An indecomposable flow $f$ on a signed graph $Σ$ is a nontrivial integral flow that cannot be decomposed into $f=f_1+f_2$, where $f_1,f_2$ are nontrivial integral flows having the same sign (both $\geq 0$ or both $\leq 0$) at each edge of $Σ$. This paper is to classify indecomposable flows into characteristic vectors of circuits and Eulerian cycle-trees --- a class of signed graphs having a kind of tree structure in which all cycles can be viewed as vertices of a tree. Moreover, each indecomposable flow other than circuit characteristic vectors can be further decomposed into a sum of certain h…
A signed graph is a graph with a positive or negative sign on each edge. Regarding each edge as two half edges, an orientation of a signed graph is an assignment of a direction to each of its half edges such that the two half edges of a positive edge receive the same direction and that of a negative edge receive opposite directions. A signed graph with such an orientation is called a bidirected graph. A nowhere-zero $k$-flow of a bidirected graph is an assignment of an integer from $\{-(k-1), \ldots, -1, 1, \ldots, (k-1)\}$ to each of its half edges such that Kirchhoff's law is respected, that…
On the foundations of signed graphs I: chain groups, frame matroid, and bivariate flow polynomial
本文研究含外边符号图的链群、流群等代数结构，提出双变量流多项式，并揭示其框架拟阵与单变量流多项式缺失之谜。
In this paper the concept of circular $r$-flows in a mono-directed signed graph $(G, σ)$ is introduced. That is a pair $(D, f)$, where $D$ is an orientation on $G$ and $f: E(G)\to (-r,r)$ satisfies that $|f(e)|\in [1, r-1]$ for each positive edge $e$ and $|f(e)|\in [0, \frac{r}{2}-1]\cup [\frac{r}{2}+1, r)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. The circular flow index of a signed graph $(G, σ)$ with no positive bridge, denoted $Φ_c(G,σ)$, is the minimum $r$ such that $(G, σ)$ admits a circular $r$-flow. This is the dual notion of circular c…
The presented paper studies the flow number $F(G,σ)$ of flow-admissible signed graphs $(G,σ)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,σ)$ there is a set ${\cal G}(G,σ)$ of cubic graphs such that $F(G, σ) \leq \min \{F(H,σ_H) : (H,σ_H) \in {\cal G}(G)\}$. We prove that $F(G,σ) \leq 6$ if $(G,σ)$ contains a bridge and $F(G,σ) \leq 7$ in general. We prove better bounds, if there is an element $(H,σ_H)$ of ${\cal G}(G,σ)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,σ) \leq 4$ and the b…
Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $Γ$ of order $n$, the number of nowhere-zero $Γ$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $Γ$, let $ε_2(Γ)$ be the largest integer $d$ so that $Γ$ has a subgroup isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of nowhere-zero $Γ$-flows in $G$…
A nowhere-zero $k$-flow on a graph $Γ$ is a mapping from the edges of $Γ$ to the set $\{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ$ such that, in any fixed orientation of $Γ$, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an \emph{integral flow polynomial} that counts nowhere-zero $k$-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting …
In contrast to ordinary graphs, the number of the nowhere-zero group-flows in a signed graph may vary with different groups, even if the groups have the same order. In fact, for a signed graph $G$ and non-negative integer $d$, it was shown that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $Γ$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $Γ$ of order $k$ with $ε(Γ)=d$, where $ε(Γ)$ is the largest integer $d$ for which $Γ$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We focus on the combinatorial structure of $Γ$-flows in a signed gra…
We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula $\prod_{k=1}^{n-2} Cat(k)$ for the type $A_n…
Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$. Minimum $T$-joins have many applications in combinatorial optimizations. In this paper, we show that a minimum $T$-join of a connected graph $G$ has at most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is the maximum bidegeless subgraph of $G$. Further, we are able to use this result to show that every flow-admissible signed graph $(G,σ)$…
We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the …
The NP-hard problem of correlation clustering is to partition a signed graph such that the number of conflicts between the partition and the signature of the graph is minimized. This paper studies graph signatures that allow the optimal partition to be found efficiently. We define the class of flow-partitionable signed graphs, which have the property that the standard linear programming relaxation based on so-called cycle inequalities is tight. In other words, flow-partitionable signed graphs satisfy an exact max-multiflow-min-multicut relation in the associated instances of minimum multicut. …
Integer flows on triangularly connected signed graphs
本文将Fan等人关于普通三角连通图允许无处零3-流或4-流的结论推广至符号图，证明每个允许流的三角连通符号图允许无处零4-流当且仅当它不是与特定符号关联的轮W5，且该结果具有尖锐性。
A Linear Bound on the Rich Flow Number for Graphs with a Given Maximum Degree
本文研究了最大度数给定的图的丰富流数，证明了其线性上界为(264Δ-445)。
We study the flow spectrum ${\cal S}(G)$ and the integer flow spectrum $\overline{\cal S}(G)$ of signed $(2t+1)$-regular graphs. We show that if $r \in {\cal S}(G)$, then $r = 2+\frac{1}{t}$ or $r \geq 2 + \frac{2}{2t-1}$. Furthermore, $2 + \frac{1}{t} \in {\cal S}(G)$ if and only if $G$ has a $t$-factor. If $G$ has a 1-factor, then $3 \in \overline{\cal S}(G)$, and for every $t \geq 2$, there is a signed $(2t+1)$-regular graph $(H,σ)$ with $ 3 \in \overline{\cal S}(H)$ and $H$ does not have a 1-factor. If $G$ $(\not = K_2^3)$ is a cubic graph which has a 1-factor, then $\{3,4\} \subseteq {\ca…
Signed circuit $6$-covers of signed $K_4$-minor-free graphs
本文针对符号图的符号回路覆盖问题，证明了Fan猜想对无K₄子式的符号图成立，即每个可覆盖的无K₄子式符号图都具有符号回路6-覆盖，并说明6-覆盖是最优可能的。
This paper is a sequel of our preceding paper (N. Kita: Constructive characterization for signed analogue of critical graphs I: Principal classes of radials and semiradials. arXiv preprint, arXiv:2001.00083, 2019). In the preceding paper, the concepts of radials and semiradials are introduced, and constructive characterizations for five principal classes of radials and semiradials are provided. Radials are a common analogue of critical graphs from matching theory and a class of directed graphs called flowgraphs, whereas semiradials are a relaxed concept of radials. Five classes of radials and …
A shortest circuit cover ${\cal F}$ of a bridgeless graph $G$ is a family of circuits that covers every edge of $G$ and is of minimum total length. The total length of a shortest circuit cover ${\cal F}$ of $G$ is denoted by $SCC(G)$. For ordinary graphs (graphs without sign), the subject of shortest circuit cover is closely related to some mainstream areas, such as, Tutte's integer flow theory, circuit double cover conjecture, Fulkerson conjecture, and others. For signed graphs $G$, it is proved recently by Máčajová, Raspaud, Rollová and Škoviera that $SCC(G) \leq 11|E|$ if $G$ is s-bridgeles…
Theoretical results for Perfect Location signed Roman domination problem
本文研究图论中的完美定位符号罗马控制（PLSRD）问题，给出多种经典图类的PLSRD数精确值，并推导3-正则图下界与花斯诺克图上界，揭示约束与图结构的关联。
Enumeration of one-cycle signed chord diagrams and their duals for optimal Morse flows on closed surfaces
该研究将闭曲面上最优Morse流的分类构造算法化，提出枚举单圈带符号弦图及对偶的算法，经多类数据验证，可分组等价类并识别自对偶类。
The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. For a fixed signed graph $\widehat{H}$, the list homomorphism problem asks whether an input signed graph $\widehat{G}$ with lists $L(v) \subseteq V(\widehat{H}), v \in V(\widehat{G}),$ admits a homomorphism $f$ to $\widehat{H}$ with all $f(v) \in L(v), v \in …
We prove that an eulerian graph $G$ admits a decomposition into $k$ closed trails of odd length if and only if and it contains at least $k$ pairwise edge-disjoint odd circuits and $k\equiv |E(G)|\pmod{2}$. We conjecture that a connected $2d$-regular graph of odd order with $d\ge 1$ admits a decomposition into $d$ odd closed trails sharing a common vertex and verify the conjecture for $d\le 3$. The case $d=3$ is crucial for determining the flow number of a signed eulerian graph which is treated in a separate paper (arXiv:1408.1703v2). The proof of our conjecture for $d=3$ is surprisingly diffic…
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