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 bound is tight. If $H$ is 3-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,σ) \leq 6$. Furthermore, if Tutte's 5-Flow Conjecture is true, then $(G,σ)$ admits a nowhere-zero 6-flow endowed with some strong properties.
A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles. Here I describe several questions regarding negative circles and their cousins the positive circles. Topics include incidence between signed circles and edges or vertices, characterizing signed graphs with special circle properties, counting negative circles, signed-circle packing and covering, signed circles and eigenvalues, and directed cycles in signed di…
We examine the conditions under which a signed graph contains an edge or a vertex that is contained in a unique negative circle or a unique positive circle. For an edge in a unique signed circle, the positive and negative case require the same structure on the underlying graph, but the requirements on the signature are different. We characterize the structure of the underlying graph necessary to support such an edge in terms of bridges of a circle. We then use the results from the edge version of the problem to help solve the vertex version.
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 $2+\frac 1k$.
I propose that most problems about circles (cycles, circuits) in ordinary graphs that have odd or even length find their proper setting in the theory of signed graphs, where each edge has a sign, $+$ or $-$. Even-circle and odd-circle problems correspond to questions about positive and negative circles in signed graphs. (The sign of a circle is the product of its edge signs.) I outline questions about circles in signed graphs, that seem natural and potentially important.
The closed neighborhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having an end-vertex in common with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1,1\}$. If $\sum_{x\in{N[e]}}f(x)\geq 1$ for each edge $e \in E(G)$, then $f$ is called a signed edge dominating function of $G$. The signed edge domination number of $G$ is the minimum weight of a signed edge dominating function of $G$. In this paper, we find the signed edge domination number of the complete tripartite graph $K_{m,n,p}$, where $1\leq m\leq n$ and $p\geq m+n$.…
A signed graph is a graph where the edges are assigned labels of either "$+$" or "$-$". The sign of a cycle in the graph is the product of the signs of its edges. We equip each signed complete graph with a vector whose entries are the number of negative $k$-cycles for $k\in\{3,\dots,n\}$. These vectors generate an affine subspace of $\mathbb{R}^{n-2}$. We prove that this subspace is all of $\mathbb{R}^{n-2}$.
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.
We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by Δ + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.
A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V) is called its signed degree set. In this paper, we prove that every set of integers is the signed degree set of some connected signed bipartite graph. \medskip \medskip
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…
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 give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than -2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than -3.
A signed graph $G$ is a graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex $v$ in a signed graph, denoted by $sdeg(v)$, is the number of positive edges incident to $v$ subtracted by the number of negative edges incident to $v$. Finally, we say $G$ realizes the set $D$ if: $$ D = \{sdeg(v) \text{ : } v\in V(G) \}. $$ The topic of signed degree sets and signed degree sequences has been studied from many directions. In this paper, we study properties needed for signed trees to have a given signed degree set. We start by proving that $D$ is …
Signed list edge coloring in graphs of bounded treewidth
Yan’an University（延安大学）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院与分析数学及应用重点实验室（教育部））；School of Marine Science and Technology, Northwestern Polytechnical University（西北工业大学海洋科学与技术学院）
An edge uv in a graph Γ is directionally 2-signed (or, (2,d)-signed) by an ordered pair (a,b), a,b in {+,-}, if the label l(uv) = (a,b) from u to v, and l(vu) = (b,a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sampathkumar by proving that the signed graph is antibalanced (all even cycles and only even cycles have positive edge sign product) if, and only if, in the bidirected graph, after suitable reorientation of edg…
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 graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,...,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for …
A "signed graph" is a graph $Γ$ where the edges are assigned sign labels, either "$+$" or "$-$". The sign of a cycle is the product of the signs of its edges. Let $\mathrm{SpecC}(Γ)$ denote the list of lengths of cycles in $Γ$. We equip each signed graph with a vector whose entries are the numbers of negative $k$-cycles for $k\in\mathrm{SpecC}(Γ)$. These vectors generate a subspace of $\mathbb R^{\mathrm{SpecC}(Γ)}$. Using matchings with a strong permutability property, we provide lower bounds on the dimension of this space; in particular, we show for complete graphs, complete bipartite graphs…
Let $Γ=(K_{n},H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. The index of $Γ$ is the largest eigenvalue of its adjacency matrix. In this paper we study the index of $Γ$ when $H$ is a unicyclic graph. We show that among all signed complete graphs of order $n>5$ whose negative edges induce a unicyclic graph of order $k$ and maximizes the index, the negative edges induce a triangle with all remaining vertices being pendant at the same vertex of the triangle.
A signed graph $(G, Σ)$ is a graph $G$ and a subset $Σ$ of its edges which corresponds to an assignment of signs to the edges: edges in $Σ$ are negative while edges not in $Σ$ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of $(G,Σ)$ to $(H,Π)$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipar…
In a signed graph each edge has a sign, $+1$ or $-1$. We introduce in the present paper a new definition of connection in a signed graph by the existence of both positive and negative chains between vertices. We prove some results and properties of this definition, such as sign components, sign articulation vertices, and sign isthmi, and we compare them to corresponding graph and signed-graphic matroid properties. We apply our results to signed graphs without positive cycles. For signed graphs in which every edge is negative our properties become parity properties.
A signed graph $Γ(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $Γ(G)$. The energy of a signed graph $Γ(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(Γ(G))$ of $Γ(G)$. The random signed graph model $\mathcal{G}_n(p, q)$ is defined as follows: Let $p, q \ge 0$ be fixed, $0 \le p+q \le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edge…
We introduce joins of signed graphs and explore the chromatic number of the all-positive and all-negative joins. We prove an analogue to the theorem that the chromatic number of the join of two graphs equals the sum of their chromatic numbers. Given two signed graphs, the chromatic number of the all-positive and all-negative join is usually less than the sum of their chromatic numbers, by an amount that depends on the new concept of deficiency of a signed-graph coloration.
Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least $-\frac{n^2}{25}$. Also we provide an example of a weighted graph with described properties and the sum of weights $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$. The previous best known bounds were $-\frac{n^2}{16}$ and $-(1+o(1))\frac{n^2}{54}$ respectively. We show that the constant $-1/54$ is optimal under some additional conditions.
A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph $(G,σ)$ to a signed graph $(H, π)$ to be a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ which preserves incidence, adjacency and the signs of closed walks. In this work we first give a characterization of the s…
Colouring negative exact-distance graphs of signed graphs
Zhejiang Normal University（浙江师范大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息学研究所）；Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557)（分析与数学社会中心）；Instituto de Ingeniería Matemática-CIMFAV（数学工程研究所-CIMFAV）；Universidad de Valparaíso（瓦尔帕莱索大学）；Computer Science Institute of Charles University（查理大学计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. The adjacency matrix of $Γ=(G, σ)$ is a square matrix $A(Γ)=A(G, σ)=\left(a_{i j}^σ\right)$, where $a_{i j}^σ=σ\left(v_{i} v_{j}\right) a_{i j}$. In this paper, we determine the eigenvalues of the signed complete bipartite graphs. Let $(K_{p, q},σ)$, $p\leq q$, be a signed complete bipartite graph with bipartition $(U_p, V_q)$, where $U_p=\{u_1,u_2,\ldots,u_p\}$ and $V_q=\{v_1,v_2,\ldots,v_q\}$. Let $(K_{p, q},σ)[U_r\cup V_s]$, $r\leq p$ and $s\leq q $, be an induced signed subgraph on minimum vertices $r+s$, …
A signed graph has edge signs. A gain graph has oriented edge gains drawn from a group. We define the combination of the two for the abelian case, in which each oriented edge of a signed graph has a gain from an abelian group, concentrating on the case of the additive group of a field. We develop the elementary graph properties, the associated matroid, and the vector and hyperplanar representations.
A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$ satisfies $\f(v)\not= \sg(vw)\f(w)$, where $\sg(vw)$ is the sign of the edge $vw$. For an integer $h\geq 0$, let $\Ga_{2h}=\{\pm1,\pm2, \ldots, \pm h\}$ and $\Ga_{2h+1}=\Ga_{2h} \cup \{0\}$. Following \cite{MaRS2015}, the signed chromatic number $\scn(G)$ of $G$ is the least integer $k$ such that $G$ admits a vertex coloring $\f$ with ${\rm im}(\f)\subseteq \Ga_k$…
A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed $K_3$'s, three signed $K_4$'s, and seven signed $K_5$'s. In this paper, we prove that there are sixteen signed $K_6$'s upto switching ismomorphism.
Recently Naserasr, Sopena, and Zaslavsky [R. Naserasr, É. Sopena, T. Zaslavsky,Homomorphisms of signed graphs: An update, arXiv: 1909.05982v1 [math.CO] 12 Sep 2019.] published a report on closed walks in signed graphs. They gave a characterization of the sets of closed walks in a graph $G$ which corespond to the set of negative walks in some signed graph on $G$. In this note we show that their characterization is not valid and give a new characterization.
Signed shifts are generalizations of the shift map in which, interpreted as a map from the unit interval to itself sending x to the fractional part of Nx, some slopes are allowed to be negative. Permutations realized by the relative order of the elements in the orbits of these maps have been studied recently by Amigo, Archer and Elizalde. In this paper, we give a complete characterization of the permutations (also called patterns) realized by signed shifts. In the case of the negative shift, which is the signed shift having only negative slopes, we use the characterization to give an exact enu…
A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs.
An oriented hypergraph is a hypergraph together with an incidence orientation such that each edge-vertex incidence is given a label of $+1$ or $-1$. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors, and provide …
A signed graph $(G,Σ)$ is a graph $G$ together with a set $Σ\subseteq E(G)$ of negative edges. A circuit is positive if the product of the signs of its edges is positive. A signed graph $(G,Σ)$ is balanced if all its circuits are positive. The frustration index $l(G,Σ)$ is the minimum cardinality of a set $E \subseteq E(G)$ such that $(G-E,Σ-E)$ is balanced, and $(G,Σ)$ is $k$-critical if $l(G,Σ) = k$ and $l(G-e, Σ- e)<k$, for every $e \in E(G)$. We study decomposition and subdivision of critical signed graphs and completely determine the set of $t$-critical signed graphs, for $t \leq 2$. Crit…
A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,\ldots,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges of $E$ even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are adjacent and connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are adjacent and connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacen…
The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.
The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede's relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede's theorem as well as a structural description of line-consistent signed graphs.
A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V=\{1,\ldots,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd edges and the other edges of $E$ even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ real matrices $A=[a_{i,j}]$ such that if $a_{i,j} < 0$, then there must be an even edge connecting $i$ and $j$; if $a_{i,j} > 0$, then there must be an odd edge connecting $i$ and $j$; and if $a_{i,j} = 0$, then either there must be an odd edge and an even edge connecting $i$ and $j$, or there are no edges connect…
