A signed graph is a simple graph with two types of edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$.
A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ such that, after switching any number of the vertices of $G$, $φ$ maps every edge of $G$ to an edge of the same type in $H$. The chromatic number $χ_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$.
We show that the chromatic number of signed triangular grids is at most 10 and the chromatic number of signed hexagonal grids is at most 4.
A signed graph $(G, σ)$ is a graph $G$ along with a function $σ: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph $(G, σ)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, π)$ to which $(G, σ)$ admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention …
A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph $G$ to a 2-edge-colored graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ that maps every edge in $G$ to an edge of the same type in $H$. Switching a vertex $v$ of a 2-edge-colored or signed graph corresponds to changing the type of each edge incident to $v$. There is a homomorphism from the signed graph $G$ to the signed graph $H$ if after switching some subset of the vertices of $G$ there is a 2-edge-colored homomorphism from $G$ to $H$. The chromatic number of a 2-e…
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arran…
In 2015, Matthias Beck and his team developed a computer program in SAGE which efficiently determines the number of signed proper $k$-colorings for a given signed graph. In this article, we determine the number of different signatures on Book graph up to switching isomorphisms. We also find a recursive formula of the signed chromatic polynomials of signed Book graphs.
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.
A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ such that, after switching some of the vertices of $G$, $φ$ maps every edge of $G$ to an edge of $H$ of the same type. The chromatic number $χ_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$. The maximum average degree $mad(G)$ …
Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs and that they could be told apart by their chromatic polynomials, by showing that the latter give distinct results when evaluated at 3. He conjectured that the six different signed Petersen graphs also have distinct zero-free chromatic polynomials, and that both types of chromatic polynomials have distinct evaluations at \emph{any} positive integer. We developed and executed a computer program (running in {\tt SAGE}) that efficiently determines the number of proper $k$-colorings for a giv…
Signed graphs are studied since the middle of the last century. Recently, the notion of homomorphism of signed graphs has been introduced since this notion captures a number of well known conjectures which can be reformulated using the definitions of signed homomorphism. In this paper, we introduce and study the properties of some target graphs for signed homomorphism. Using these properties, we obtain upper bounds on the signed chromatic numbers of graphs with bounded acyclic chromatic number and of signed planar graphs with given girth.
For $m \geq 3$ and $n \geq 1$, the $m$-cycle book graph $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $ϕ\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $ϕ(u)$ is different from the colour $σ(uv)ϕ(v)$, where is $σ(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs…
A signed graph is a pair $(G, σ)$, where $G$ is a graph and $σ: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, σ)$ a circular $r$-coloring of $(G, σ)$ is an assignment $ψ$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $σ(e)=+$, then $ψ(u)$ and $ψ(v)$ have distance at least $1$, and if $σ(e)=-$, then $ψ(v)$ and the antipodal of $ψ(u)$ have distan…
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $χ_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|χ_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplan…
We study Dohmen--Pönitz--Tittmann's bivariate chromatic polynomial $c_Γ(k,l)$ which counts all $(k+l)$-colorings of a graph $Γ$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_Γ(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_Γ(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynom…
We introduce the concept of deficiency in signed graphs. The deficiency of a coloration is the number of unused colors. We classify the deficiency of 2-chromatic graphs. There are four decision problems about the minimum and maximum deficiency of a 3-chromatic signed graph. We answer two of them with a polynomial-time algorithm for deciding the maximum deficiency of a 3-chromatic signed graph.
We extend the work of Hanlon on the chromatic polynomial of an unlabeled graph to define the unlabeled chromatic polynomial of an unlabeled signed graph. Explicit formulas are presented for labeled and unlabeled signed chromatic polynomials as summations over distinguished order-ideals of the signed partition lattice. We also define the quotient of a signed graph by a signed permutation, and show that its signed graphic arrangement is closely related to an induced arrangement on a distinguished subspace. Lastly, a formula for the number of unlabeled acyclic orientations of a signed graph is pr…
Let $Σ$ be a signed graph where two edges joining the same pair of vertices with opposite signs are allowed. The zero-free chromatic number $χ^*(Σ)$ of $Σ$ is the minimum even integer $2k$ such that $G$ admits a proper coloring $f\colon\,V(Σ)\mapsto \{\pm 1,\pm 2,\ldots,\pm k\}$. The zero-free list chromatic number $χ^*_l(Σ)$ is the list version of zero-free chromatic number. $Σ$ is called zero-free chromatic-choosable if $χ^*_l(Σ)=χ^*(Σ)$. We show that if $Σ$ has at most $χ^*(Σ)+1$ vertices then $Σ$ is zero-free chromatic-choosable. This result strengthens Noel-Reed-Wu Theorem which states th…
The chromatic number $χ((G,σ))$ of a signed graph $(G,σ)$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= σ(e) c(w)$ for every edge $e = vw$. Let $Σ(G)$ be the set of all signatures of $G$. We study the chromatic spectrum $Σ_χ(G) = \{χ((G,σ))\colon\ σ\in Σ(G)\}$ of $(G,σ)$. Let $M_χ(G) = \max\{χ((G,σ))\colon\ σ\in Σ(G)\}$, and $m_χ(G) = \min\{χ((G,σ))\colon\ σ\in Σ(G)\}$. We show that $Σ_χ(G) = \{k : m_χ(G) \leq k \leq M_χ(G)\}$. We also prove some basic facts for critical graphs. Analogous results are obtained for a notion of …
A signed graph $ (G, Σ)$ is a graph positive and negative ($Σ$ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, Σ)$ is to switch the signs of the edges incident to $v$. If one can obtain $ (G, Σ')$ by re-signing some vertices of $ (G, Σ)$, then $ (G, Σ) \equiv (G, Σ')$. A signed graphs $ (G, Σ)$ admits an homomorphism to $ (H, Λ)$ if there is a sign preserving vertex mapping from $(G,Σ')$ to $(H, Λ)$ for some $ (G, Σ) \equiv (G, Σ')$. The signed chromatic number $χ_{s}( (G, Σ))$ of the signed graph $(G, Σ)$ is the minimum order (number of vertices) of a signe…
Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $χ_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number $χ$. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are si…
The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been investigated in the literature, including variants for modifications of proper colourings. These variants include, notably, the achromatic number of graphs, which is the maximum number of colours required to colour the vertices of graphs properly so that each possible combination of distinct colours is assigned along some edge. The behaviours of this parameter have l…
Balanced-chromatic number and Hadwiger-like conjectures
Universidad de Valparaíso（瓦尔帕莱索大学）；Millennium Nucleus for Social Data Science (SODAS)（社会数据科学千年核（SODAS））；Auburn University（奥本大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息研究与研究所）；École Normale Supérieure, Paris（巴黎高等师范学院）
The notion of the circular coloring of signed graphs is a recent one that simultaneously extends both notions of the circular coloring of graphs and $0$-free coloring of signed graphs. A circular $r$-coloring of a signed graph $(G, σ)$ is to assign points of a circle of circumference $r$, $r\geq 2$, to the vertices of $G$ such that vertices connected by a positive edge are at circular distance at least $1$ and vertices connected by a negative edge are at circular distance at most $\frac{r}{2}-1$. The infimum of all $r$ for which $(G, σ)$ admits a circular $r$-coloring is said to be the circula…
There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks'-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the under…
Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.
In this paper, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper bound on its chromatic number.
A circular $r$-coloring of a signed graph $(G, σ)$ is an assignment $ϕ$ of points of a circle $C_r$ of circumference $r$ to the vertices of $(G, σ)$ such that for each positive edge $uv$ of $(G, σ)$ the distance of $ϕ(v)$ and $ϕ(v)$ is at least 1 and for each negative edge $uv$ the distance of $ϕ(u)$ from the antipodal of $ϕ(v)$ is at least 1. The circular chromatic number of $(G, σ)$, denoted $χ_c(G, σ)$, is the infimum of $r$ such that $(G, σ)$ admits a circular $r$-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed $d$-d…
In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed…
Although it has recently been proved that the packing chromatic number is unbounded on the class of subcubic graphs, there exists subclasses in which the packing chromatic number is finite (and small). These subclasses include subcubic trees, base-3 Sierpi{ń}ski graphs and hexagonal lattices.In this paper we are interested in the packing chromatic number of subcubic outerplanar graphs. We provide asymptotic bounds depending on structural properties of the outerplanar graphs and determine sharper bounds for some classes of subcubic outerplanar graphs.
Vertex coloring of a graph $G$ with $n$-colors can be equivalently thought to be a graph homomorphism (edge preserving vertex mapping) of $G$ to the complete graph $K_n$ of order $n$. So, in that sense, the chromatic number $χ(G)$ of $G$ will be the order of the smallest complete graph to which $G$ admits a homomorphism to. As every graph, which is not a complete graph, admits a homomorphism to a smaller complete graph, we can redefine the chromatic number $χ(G)$ of $G$ to be the order of the smallest graph to which $G$ admits a homomorphism to. Of course, such a smallest graph must be a compl…
A signified graph is a pair $(G, Σ)$ where $G$ is a graph, and $Σ$ is a set of edges marked with '$-$'. Other edges are marked with '$+$'. A signified coloring of the signified graph $(G, Σ)$ is a homomorphism into a signified graph $(H, Δ)$. The signified chromatic number of the signified graph $(G, Σ)$ is the minimum order of $H$. In this paper we show that for every 2-dimensional grid $(G, Σ)$ there exists homomorphism from $(G, Σ)$ into the signed Paley graphs $SP_9$. Hence signified chromatic number of the signified grids is at most 9. This improves upper bound on this number obtained rec…
The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary clique problem is in P. We also study the complexity of further related problems.
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）
The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of the Hajós theorem, emphasizing the role of all-positive complete graphs played in the class of signed graphs, as played in the class of unsigned graphs.
An $i$-packing in a graph $G$ is a set of vertices at pairwise distance greater than $i$. For a nondecreasing sequence of integers $S=(s\_{1},s\_{2},\ldots)$, the $S$-packing chromatic number of a graph $G$ is the least integer $k$ such that there exists a coloring of $G$ into $k$ colors where each set of vertices colored $i$, $i=1,\ldots, k$, is an $s\_i$-packing. This paper describes various subdivisions of an $i$-packing into $j$-packings ($j\textgreater{}i$) for the hexagonal, square and triangular lattices. These results allow us to bound the $S$-packing chromatic number for these graphs,…
Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process we develop new properties of signed graphs, especially of their switching automorphism groups.
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…
In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated $(3+1)$-free conjecture. Our method is to follow the generalization of the chromatic symmetric functions by Gebhard and Sagan to symmetric functions in non-commuting variables. These functions satisfy a deletion-contraction property unlike the chromatic symmetric function in commuting variables. We do this by proving a new signed combinatorial formula for \emph{…
In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an …
