Circular chromatic number of signed graphs

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph and $\sigma: E(G) \to \{+, -\}$ 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 $(G, \sigma)$ a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of M\'{a}\v{c}ajov\'{a}, Raspaud, and \v{S}koviera.


Introduction
Assume r ≥ 1 is a real number. We denote by C r the circle of circumference r, obtained from the interval [0, r] by identifying 0 and r. Points in C r are real numbers from [0, r). For two points x, y on C r , the distance between x and y on C r , denoted by d (mod r) (x, y), is the length of the shorter arc of C r connecting x and y. Given two real numbers a and b, the interval [a, b] on C r is a closed interval of C r in clockwise orientation of the circle whose first point is a(mod r) and whose end point is b(mod r). the intervals ϕ(u) and ϕ(v) do not intersect and for a negative edge uv, the intervals ϕ(u) and ϕ(v) do not intersect.
Observe that if (G, σ) has no edge, then χ c (G, σ) = 1, and if (G, σ) has an edge, either positive or negative, then (G, σ) is not circular r-colorable for r < 2. As graphs with no edge are not interesting, in the remainder of the paper, we always assume that r ≥ 2.
It follows from the definition that for any graph G, χ c (G, +) = χ c (G). So the circular chromatic number of a signed graph is indeed a generalization of the circular chromatic number of a graph. The circular chromatic number of a graph is a refinement of its chromatic number: for any positive integer k, a graph G is circular k-colorable if and only if G is k-colorable. The same is also true for the chromatic number of signed graphs defined based on the notion of 0-free coloring define by Zaslavsky [22]. Proof. Assume f : V (G) → {±1, ±2, . . . , ±k} is any mapping. Let It is straightforward to verify that g is a circular 2k-coloring of (G, σ) if and only if f is a 0-free 2k-coloring of (G, σ).
The number of colors used in the 0-free coloring is always even. There have been several attempts to introduce an analogue coloring which uses an odd number of colors. The term 0-free indeed identifies this coloring from a similar coloring where 0 is added to the set of colors and the set of vertices colored with 0 induces an independent set. To be precise, a (2k + 1)-coloring of a signed graph uses colors {0, ±1, . . . , ±k}, and the constraint is still the same: for any edge e = uv of G, f (u) = σ(e)f (v). In a (2k + 1)-coloring of a signed graph, the color 0 is different from the other colors. The antipodal of 0 is 0 itself. The set of vertices of color 0 is an independent set of G, and for every other color i, vertices colored by color i may be joined by negative edges. In some sense, circular coloring of signed graph provides a more natural generalization of 0-free coloring to colorings of signed graphs with an odd number of colors, where the colors are symmetric.
In this paper, we shall study basic properties of circular coloring of signed graphs. We shall explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and prove that for any graph G, χ c (G) ≤ χ s c (G) ≤ 2χ c (G). We prove that the upper bound is tight even when restricted to graphs of arbitrary large girth or bipartite planar graphs. Furthermore, we construct a signed planar simple graph whose circular chromatic number is 4 + 2 3 . Máčajová, Raspaud, and Škoviera [13] conjectured that every signed planar simple graph is 4-colorable. By Proposition 2.10, this is equivalent to say that χ s c (G) ≤ 4 for every planar graph. Kardos and Narboni [9] refuted this conjecture by constructing a non-4-colorable signed planar graph. Our construction improves on the example of Kardos and Narboni. Thus we show that the supremum of the signed chromatic number of planar graphs is between 4 + 2 3 and 6. The exact value remains an open problem.

Equivalent definitions
There are several equivalent definitions of the circular chromatic number of graphs. Some of these definitions are also extended naturally to signed graphs.
Note that for s, t ∈ [0, r), d (mod r) (s, t) = min{|s − t|, r − |s − t|}. So a circular r-coloring of a graph can be defined as follows, which is sometimes more convenient.
If r is a rational number, then in a circular r-coloring of a signed graph (G, σ), it suffices to use a finite set of colors from the interval [0, r). We may assume that r = p q , where p is even and subject to this condition p q is in its simplest form. For i ∈ {0, 1, . . . , p − 1}, let I i be the half open, half closed interval [ i q , i+1 q ) of [0, r). Then ∪ p−1 i=0 I i is a partition of [0, r). Assume f : V (G) → [0, r) is a circular r-coloring of a signed graph (G, σ). Then for each vertex v of G, let g(v) = i q if and only if f (v) ∈ I i . If e = uv is a positive edge, then 1 ≤ |f (u) − f (v)| ≤ p q − 1. This implies that 1 − 1 q < |g(u) − g(v)| < p q − 1 + 1 q . Since q|g(u) − g(v)| is an integer, we conclude that 1 ≤ |g(u) − g(v)| ≤ p q − 1. If e = uv is a negative edge, then either |g(u) − g(v)| < p 2 − 1 + 1 q or |g(u) − g(v)| > p 2 + 1 − 1 q . Since p is even, p 2 is an integer. As q|g(u) − g(v)| is an integer, we conclude that either |g(u) − g(v)| ≤ p 2 − 1 or |g(u) − g(v)| ≥ p 2 + 1. It is crucial that p be an even integer. For otherwise p 2 is not an integer, and we cannot conclude that |g(u) − g(v)| ≤ p 2 − 1 or |g(u) − g(v)| ≥ p 2 + 1. Indeed, if p is odd, then the set {0, 1 q , . . . , p−1 q } is not closed under taking antipodal points.
The above observation leads to the following equivalent definition of the circular chromatic number of signed graphs. For i, j ∈ {0, 1, . . . , p − 1}, the modulo-p distance between i and j is Given an even integer p, the antipodal color of x ∈ {0, 1, . . . , p − 1} isx = x + p 2 (mod p).
Definition 2.2. Assume p is an even integer and q ≤ p/2 is a positive integer. A (p, q)-coloring of a signed graph (G, σ) is a mapping f : V (G) → {0, 1, . . . , p − 1} such that for any positive edge uv, and for any negative edge uv, The circular chromatic number of (G, σ) is : p is an even integer and (G, σ) has a (p, q)-coloring}.
is an edge of H. It is well-known and easy to see that a graph G is k-colorable if and only if G admits a homomorphism to K k , the complete graph on k vertices. Similarly, circular chromatic number of graphs are also defined through graph homomorphism. For integers p ≥ 2q > 0, the circular clique K p;q has vertex set [p] = {0, 1, . . . , p − 1} and edge set {ij : q ≤ |i − j| ≤ p − q}. Then a circular p q -coloring of a graph G is equivalent to a homomorphism of G to K p;q . Circular chromatic number of signed graphs can also be defined through homomorphisms.
3. An edge-sign preserving homomorphism of a signed graph (G, σ) to a signed graph (H, π) is a mapping f : V (G) → V (H) such that for every positive (respectively, negative) edge uv of We write (G, σ) s.p.
−→ (H, π) if there exists an edge-sign preserving homomorphism of (G, σ) to (H, π). For integers p ≥ 2q > 0 such that p is even, the signed circular clique K s p;q has vertex set [p] = {0, 1, . . . , p − 1}, in which ij is a positive edge if and only if q ≤ |i − j| ≤ p − q and ij is a negative edge if and only if either |i − j| ≤ p 2 − q or |i − j| ≥ p 2 + q. If q = 1, then K s p;1 is also written as K s p . Note that in K s p;q , each vertex i is incident to a negative loop. When p q ≥ 4, there are parallel edges of different signs. Furthermore, the subgraph induced by all the positive edges of K s p;q is the circular clique K p;q , which is known to be of circular chromatic number p q , thus we have χ c (K s p;q ) = p q . The following lemma gives another equivalent definition of the circular chromatic number of a signed graph. Lemma 2.4. Assume (G, σ) is a signed graph, p is a positive even integer, q is a positive integer and p ≥ 2q. Then (G, σ) has a (p, q)-coloring if and only if (G, σ) s.p.
−→ K s p;q . Hence the circular chromatic number of (G, σ) is As homomorphism relation is transitive, we have the following lemma.
For a real number r ≥ 2, we can also define K s r be the infinite graph with vertex set [0, r), in which xy is a positive edge if 1 ≤ |x − y| ≤ r − 1 and xy is a negative edge if either |x − y| ≤ r 2 − 1 or |x − y| ≥ r 2 + 1. Then it follows from the definition that a signed graph (G, σ) is circular r-colorable if and only if (G, σ) admits an edge-sign preserving homomorphism to K s r . If r = p q is a rational and p is an even integer, then it follows from the definition that K s p;q is a subgraph of K s r . On the other hand, it follows from Lemma 2.4 that K s r admits an edge-sign preserving homomorphism to K s p;q . Note that if Assume (G, σ) is a signed graph. A switching at vertex v is to switch the signs of edges which are incident to v. A switching at a set A ⊂ V (G) is to switch at each vertex in A. That is equivalent to switching the signs of all edges in the edge-cut E(A, V (G) \ A). A signed graph (G, σ) is a switching of (G, σ ′ ) if it is obtained from (G, σ ′ ) by a sequence of switchings. We say (G, σ) is switching equivalent to (G, σ ′ ) if (G, σ) is a switching of (G, σ ′ ). It is easily observed that given a graph G, the relation "switching equivalent" is an equivalence class on the set of all signatures on G.
It was observed in [23] that if (G, σ) admits a 0-free 2k-coloring then every switching equivalent signed graph (G, σ ′ ) admits such a coloring: If c is a 0-free 2k-coloring of (G, σ), then after a switching at a vertex v one may change the color of v from c(v) to −c(v) to preserve the property of being a 0-free 2k-coloring. The same argument applies to circular r-coloring.
Assume (G, σ) is a signed graph and c is a (p, q)-coloring of (G, σ) (where p is even and subject to this condition p q is in its simplest form). Let A = {v : c(v) ≥ p 2 } and let (G, σ ′ ) be obtained from (G, σ) by switching at A. It follows from the proof of Proposition 2.7 that there is a (p, q)-coloring c ′ of (G, σ ′ ) Thus, in particular, Lemma 2.5 and Lemma 2.6 can be restated with a switching homomorphism in place of edge-sign preserving homomorphism.
Note that in the graphK s p , every pair of distinct vertices are joined by a positive edge and a negative edge, and moreover, each vertex i is incident to a negative loop. Thus we have the following result. Proposition 2.10. A signed graph (G, σ) is (2k, 1)-colorable (equivalently 0-free 2k-colorable) if and only if there is a set A of vertices such that after switching at A, the result is a signed graph whose positive edges induce a k-colorable graph.
In the study of circular coloring of signed graphs, switching-equivalent signed graphs are viewed as the same signed graph. The problem as which signed graphs are equivalent was first studied by Zaslavsky [23]. We define the sign of a cycle (respectively, a closed walk) in (G, σ) to be the product of the signs of the edges of the cycle (respectively, the closed walk). One may observe that a switching does not change the sign of a cycle of (G, σ). A result of Zaslavsky, fundamental in the study of signed graphs, shows that a switching equivalent class to which (G, σ) belongs to is determined by signs of all cycles of (G, σ). Thus we have the following proposition (see [18] for more details). Proposition 2.12. A signed graph (G, σ) admits a switching homomorphism to a signed graph (H, π) if and only if there is a homomorphism f from G to H such that for every closed walk W of (G, σ), W and f (W ) have the same sign.
The following lemma follows from Theorem 2.11. Lemma 2.13. A signed graph (G, σ) admits a switching homomorphism to (H, π) if and only if there is a mapping of vertices and edges of (G, σ) to the vertices and edges of (H, π) which preserves adjacencies, incidences, and signs of closed walks.
For a non-zero integer ℓ, we denote by C ℓ the cycle of length |ℓ| whose sign agrees with the sign of ℓ.
So for example C −4 is a negative cycle of length 4. Observe that the signed graphK 4k;2k−1 is obtained from C −2k by adding a negative loop at each vertex. Note that adding negative loops to a signed graph or deleting them does not affect its circular chromatic number. So we may ignore negative loops in (G, σ). However, as a target of switching homomorphism, negative loops are important, because we can map two vertices connected by a negative edge to a same vertex v, provided v is incident to a negative loop.

Some basic properties
Assume (G, σ) is a signed graph and φ : V (G) → [0, r) is a circular r-coloring of (G, σ). The partial orientation D = D φ (G, σ) of G with respect to a circular r-coloring φ is defined as follows: (u, v) is an arc of D if and only if one of the following holds: • uv is a positive edge and (φ(v) − φ(u))(mod r) = 1.
Definition 3.1. Assume (G, σ) is a signed graph and φ is a circular r-coloring of (G, σ). Arcs in D φ (G, σ) are called tight arcs of (G, σ) with respect to φ. A directed path (respectively, a directed cycle) in D φ (G, σ) is called a tight path (respectively, a tight cycle) with respect to φ. Lemma 3.2. Let (G, σ) be a signed graph and let φ be a circular r-coloring of (G, σ). If D φ (G, σ) is acyclic, then there exists an r 0 r such that (G, σ) admits an r 0 -circular coloring.
As D φ (G, σ) has no arc, it follows from the definition that there exists ǫ > 0 such that for any positive edge uv, and for any negative edge uv, Let r 0 = r 1+ǫ and let ψ : Then ψ is an r 0 -circular coloring of (G, σ).  Proof. One direction is proved in Corollary 3.3. It remains to show that if χ c (G, σ) < r, then there is a circular r-coloring φ of (G, σ) such that D φ (G, σ) is acyclic.
Assume χ c (G, σ) = r ′ < r. Let ψ : V (G) → [0, r) be a circular r ′ -coloring of (G, σ). Let φ(v) = r r ′ ψ(v). Then it is easy to verify that φ is a circular r-coloring of (G, σ) and D φ (G, σ) contains no arc (and hence is acyclic). Proposition 3.5. Any signed graph (G, σ) which is not a forest has a cycle with s positive edges and t negative edges such that χ c (G, σ) = 2(s+t) 2a+t for some non-negative integer a.
Proof. Assume χ c (G, σ) = r and ψ : V (G) → [0, r) is a circular r-coloring of (G, σ). By Lemma 3.4, Assume B consists of s positive edges and t negative edges. We view the colors as the points of a circle C r of circumference r, which is obtained from the interval [0, r] by identifying 0 and r. Assume is a positive edge, then traversing from the colors of v i , one unit along the clockwise direction of C r , we arrive at the color of v i+1 . If v i v i+1 is a negative edge, then from the color of v i , by first traversing r 2 unit along the anti-clockwise direction of C r then traversing along the clockwise direction a unit distance, we arrive at the color of v i+1 . Therefore, directed cycle B represents a total traverse along the circle C r distance s − ( r 2 − 1) · t, at end of which one must come back to the starting color. So for some integer a. Hence Since s + t ≤ |V (G)|, and r ≥ 2, given the number of vertices of G, there is a finite number of candidates for the circular chromatic number of (G, σ). Thus we have the following corollary.
Corollary 3.6. Assume (G, σ) is a signed graph on n vertices. Then χ c (G, σ) = p q for some p ≤ 2n. In particular, the infimum in the definition of χ c (G, σ) can be replaced by minimum.
It also follows from Corollary 3.6 that there is an algorithm that determines the circular circular chromatic number of a finite signed graph. Of course, determining the circular chromatic number of a signed graph is at least as hard as determining the chromatic number of a graph, and, hence, the problem is NP-hard and, unless P=NP, there is no feasible algorithm for the problem. Nevertheless, it is easy to determine whether a signed graph (G, σ) has circular chromatic number 2.
with respect to f is also a tight cycle with respect to g. However, for each edge is an arc on the circle of length r 2 along the clockwise direction.
Recall that the core of a graph G is a smallest subgraph H of G to which G admits a homomorphism.
If (G, σ) is a signed graph and H is a subgraph of G, then we denote by (H, σ) the signed subgraph of (G, σ), where σ in (H, σ) is considered to be the restriction of σ to E(H). We define the sp-core of a signed graph (G, σ) to be a smallest signed subgraph (H, σ) such that (G, σ) admits an edge-sign preserving homomorphism to (H, σ). The switching core of a signed graph (G, σ) is a smallest signed subgraph (H, σ) such that (G, σ) admits a switching homomorphism to (H, σ). That the sp-core and the switching core of a finite signed graph is unique up to isomorphism and thus the well-definiteness is shown in [17].
It follows from the definition that the switching core of (G, σ) is isomorphic to a signed subgraph of a sp-core of (G, σ).
Lemma 3.8. Assume r = p q is a rational, p is an even integer and with respect to this condition p q is in its simplest form. ThenK s p;q is the unique switching core of K s r .
Proof. SinceK s p;q is a subgraph of K s r and K s r switch −→K s p;q , it suffices to show thatK s p;q is a switching core, i.e., it is not switching homomorphic to any of its proper signed subgraphs.
Assume to the contrary that there is a switching homomorphism ofK s p;q to a proper signed subgraph, say (H, σ). As (H, σ) switch −→K s p;q andK s p;q switch −→ (H, σ), we have χ c (H, σ) = χ c (K s p;q ) = p q . Let φ be a circular p q -coloring of (H, σ). By Corollary 3.3, there is a tight cycle C with respect to φ. Assume the length of C is l. Since p q is in its simplest form, beside a possible factor of 2, we consider two cases: q is odd, then p | 2l, which implies that l ≥ p 2 ; or q is even then p 2 must be an odd number, thus , which is a proper subgraph ofK s p;q , a contradiction.
Lemma 3.9. Assume r = p q is a rational, p is an even integer and with respect to this condition p q is in its simplest form. Then K s p;q is the unique sp-core of K s r .
Proof. As K s r s.p.
−→ K s p;q , it is enough to prove that K s p;q is a sp-core. Let (H, σ) be the sp-core of K s p;q which is a proper subgraph and let ϕ be an edge-sign preserving homomorphism of K s p;q to (H, σ). Since any edge-sign preserving homomorphism is, in particular, a switching homomorphism and by Lemma 3.8, K s p;q is a subgraph of (H, σ). Observe that for each vertex u ofK s p;q there are two corresponding vertices u 1 and u 2 of K s p;q such that a switching at u 1 gives u 2 . Furthermore, there exists a positive edge and for any other vertex v ofK s p;q , as otherwise we have an edge-sign preserving homomorphism ofK s p;q to its proper subgraph by mapping u to v. It is a contradiction.

Circular chromatic number vs. signed circular chromatic number
The following lemma follows from the definitions.  For a graph G and an arbitrary signature σ, with the definition of (G ′ , τ ) given in the previous As adding or deleting negative loops does not affect the circular chromatic number, the signed graph (G, σ) obtained from K p;q by replacing each edge with a pair of positive and negative edges has circular chromatic number 2p q . So Corollary 4.3 is tight. However, this signed graph has girth 2, i.e., has parallel edges. The following result shows that the bound in Corollary 4.3 is also tight for graphs of large girth. Theorem 4.4. For any integers k, g ≥ 2, for any ǫ > 0, there is a graph G of girth at least g satisfying that χ(G) = k and χ s c (G) > 2k − ǫ. The proof of Theorem 4.4 uses the concept of augmented tree introduced in [1]. A complete k-ary tree is a rooted tree in which each non-leaf vertex has k children and all the leaves are of the same level (the level of a vertex v is the distance from v to the root). For a leaf v of T , let P v be the unique path in T from the root to v. Vertices in P v − {v} are ancestors of v. An q-augmented k-ary tree is obtained from a complete k-ary tree by adding, for each leaf v, q edges connecting v to q of its ancestors.
These q edges are called the augmenting edges from v. For positive integers k, q, g, a (k, q, g)-graph is a q-augmented k-ary tree which is bipartite and has girth at least g. The following result was proved in [1].
Assume T is a complete k-ary tree. A standard labeling of the edges of T is a labeling φ of the edges of T such that for each non-leaf vertex v, for each i ∈ {1, 2, . . . , k}, there is one edge from v to one of its child labeled by i. Given a k-coloring f : Proof of Theorem 4.4 Assume k, g ≥ 2 are integers. We shall prove that for any integer p, there is a graph G for which the followings hold: 1. G has girth at least g and chromatic number at most k.
Let H be a (2kp, k, 2kg)-graph with underline tree T . Let φ be a standard 2kp-labeling of the edges of T . For v ∈ V (T ), denote by ℓ(v) the level of v, i.e., the distance from v to the root vertex in T . Let The addition above are carried out modulo 2kp.
Let L be the set of leaves of T . For each v ∈ L, we define one edge e v on V (T ) as follows: Let (G, σ) be the signed graph with vertex set V (T ) and with edge set {e v : v ∈ L}, where the signs of the edges are defined as above. We shall show that (G, σ) has the desired properties.
First observe that θ is a proper k-coloring of G. So G has chromatic number at most k.
Next we show that G has girth at least g. For each edge Then B v has length at most 2k. If C is a cycle in G, then replace each edge e v of C by the path B v , we obtain a cycle in H. As H has girth at least 2kg, we conclude that C has length at least g and hence G has girth at least g.
This is in contrary to the assumption that f is a circular (2kp, p + 1)-coloring of (G, σ).
Remark: The graph constructed above is shown to have chromatic number at most k. However, since 2kp p+1 < χ c (G, σ) ≤ 2χ(G), we conclude that χ(G) = k when p + 1 ≥ 2k. It is not known whether there is a finite k-chromatic graph of girth at least g and with χ s c (G) = 2k. Also it is unknown whether for every rational p q and integer g and any ǫ > 0, there is a graph G with χ c (G) ≤ p q and χ s The following result about circular chromatic number of critical graphs of large girth was proved in [28].
Theorem 4.6. For any integer k ≥ 3 and ǫ > 0, there is an integer g such that any k-crtical graph of girth at least g has circular chromatic number at most k − 1 + ǫ.
As a consequence of Theorem 4.6 and Corollary 4.3, we know that for any integer k ≥ 3 and ǫ > 0, there is an integer g such that any k-critical graph G of girth at least g has signed circular chromatic number at most 2k − 2 + ǫ. However, this bound is not tight. The following proposition follows from Proposition 2.10.

Signed indicator
In the study of coloring and homomorphism of graphs, using gadgets to construct new graphs from old ones is a fruitful tool. In this section, we explore the same idea for signed graph coloring. There is a subtle issue in the above definition. An edge e = xy is an unordered pair. So we can write it as e = yx as well. However, by identifying y with u and identifying x with v, the resulting signed graph is different from the one as defined above. To avoid such confusion, it is safer to first orient the edges of Ω and then replace the directed edge e with I. However, for our usage in this paper, the difference does not affect our discussion, so we just say replace the edge e with I.  Observe that for I = (Γ, u, v), Z(I, r) = ∅ if and only if χ c (Γ) ≤ r. One useful interpretation of Z(I, r) is that this is the set of possible distances (in C r ) between the two colors assigned to u and v in a circular r-coloring of Γ.
Let the sign of a path P in (G, σ) be the product of the signs of the edges of P .
Example 5.6. If Γ is a positive 2-path connecting u and v, and I = (Γ, u, v), then for any ǫ, 0 < ǫ < 1, and r = 4 − 2ǫ, If Γ ′ is a negative 2-path connecting u and v, and I ′ = (Γ ′ , u, v), then for any ǫ, 0 < ǫ < 1, and If Γ ′′ consists of a negative 2-path and a positive 2-path connecting u and v, and I ′′ = (Γ ′′ , u, v), then for any ǫ, 0 < ǫ < 1, and r = 4 − 2ǫ, Lemma 5.7. Assume I = (Γ, u, v) is a signed indicator, r ≥ 2 is a real number and for some 0 < t < r 4 . Then for any graph G, Proof. Let r ′ = r 2t . If χ c (G) ≤ r ′ and f is a circular r ′ -coloring of G, then g : V (G) → [0, r) defined as g(x) = tf (x) satisfies the condition that for any edge e = xy of G, So d (mod r) (g(x), g(y)) ∈ Z(I, r), and the mapping g can be extended to a circular r-coloring of the copy of Γ that was used to replace e. So g can be extended to a circular r-coloring of G(I).
Conversely, assume χ c (G(I)) ≤ r. Let g be a circular r-coloring of G(I). By vertex switching, we may assume that g(x) ∈ [0, r 2 ) for every vertex x of G(I). Then for any edge e = xy of G, . Then for any edge e = xy of G, A similar proof implies the following: Proof. Let ǫ = 2 r ′ +1 and r = 4 − 2ǫ. By Example 5.6, Z(I, r) = [ǫ, r 2 − ǫ]. Note that r ′ = r 2ǫ . The conclusion follows from Lemma 5.7.
We note that G(I) here is the same as S(G) defined in [17]. In [17], it is shown that by using S(G) construction and the graph homomorphism, the chromatic number of graphs are captured by switching homomorphisms of signed bipartite graphs. This corollary shows that furthermore χ c (S(G)) also determines χ c (G).
• If i is even, then Proof. We prove the lemma by induction on i. For i = 1, this is trivial and observed in Example 5.6. Assume i ≥ 2 and the lemma holds for i ′ < i.
. Proof. Let 1 2ǫ < i < 1 ǫ . Let Γ ′ i be obtained from the disjoint union of Γ 2i−1 and Γ 2i by identifying u 2i−1 in Γ 2i−1 and u 2i in Γ 2i into a single vertex u ′ i , and identifying v 2i−1 in Γ 2i−1 and v 2i in Γ 2i into a single vertex v ′ i . It follows from the construction that Γ ′ i is a signed bipartite planar simple graph.
So Γ ′ i is not circular r-colorable.

Circular chromatic number of signed graph classes
We have shown that χ s c (G) ≤ 2χ c (G) and this bound is tight even for graphs G of large girth. However, when restricted to some natural families of graphs, the upper bound can be improved.
Given a class C of signed graphs we define χ c (C) = sup{χ c (G, σ) : (G, σ) ∈ C}. In light of Corollary 4.2 and the fact that negative loops do not affect the circular chromatic number, we shall restrict to signed graphs with no negative digons and no loops, i.e., the underlying graphs are simple graphs.
We denote by • SD d the class of signed d-degenerate simple graphs, • SSP the class of signed series parallel simple graphs, • O the class of signed outer planar simple graphs, • SBP the class of signed bipartite planar simple graphs, • SP the class of signed planar simple graphs. Proof. First we show that every (G, σ) ∈ SD d admits a circular (2⌊ d 2 ⌋ + 2)-coloring. Equivalently, (G, σ) admits an edge-sign preserving homomorphism to K s 2⌊ d 2 ⌋+2 whose vertices are labelled 0, 1, . . . , 2⌊ d 2 ⌋ + 1 in a cyclic order. Recall that in K s 2⌊ d 2 ⌋+2 between any pair of vertices x i , x j there are both positive and negative edges, unless i = j or i = j + ⌊ d 2 ⌋ + 1. When i = j, there is a negative loop but no positive loop; when i = j + ⌊ d 2 ⌋ + 1, x i x j is a positive edge but not a negative edge. Thus, given a vertex u of (G, σ) and a partial mapping φ of (G, σ) to K s 2⌊ d 2 ⌋+2 , if at most d neighbors of u are already colored, then φ can be extended to u. This now can be applied on the ordering of vertices of G which is a witness of G being d-degenerate.
To prove that the upper bound is tight, we consider three cases. For d = 2, the signed graphs built in Corollary 5.12 are all 2-degenerate and the claim of this corollary is that the limit of their circular chromatic number is 4. For odd integer d, this bound is tight by considering the signed complete graphs (K d+1 , +). For even integer d ≥ 4, we now construct a d-degenerate graph G together with a signature σ such that χ c (G, σ) = d + 2.
Define a signed graph Ω d as follows. Take (K d , +) whose vertices are labelled x 1 , x 2 , . . . , x d . For each pair i, j ∈ [d] (i = j), we add a vertex y i,j and join it to x i , x j with negative edges, and to all the other x k 's with positive edges. Since each y i,j is of degree d and after removing all of them we are left with a K d , we have Ω d ∈ SD d . We claim that χ c (Ω d ) = d + 2.
Assume this is not true and ϕ is a circular r-coloring of Ω d and r < d + 2. Without loss of generality, we may assume that ϕ(x 1 ), ϕ(x 2 ), . . . , ϕ(x d ) are cyclicly ordered on C r in a clockwise orientation.
We will now show that there is no possible choice for y 1,1+ d 2 . A point between ϕ(x i ) and ϕ(x i+1 ) for . . , d − 1} is at distance less than 1 from one of the two and cannot be the color of , ϕ(x 2 )) ≥ 1, and, furthermore, (ϕ(y 1, , then the same argument shows that which is a contradiction as y 1,1+ d 2 x 1 is a negative edge. It follows from Proposition 6.1 that χ c (G, σ) ≤ 2⌊ ∆(G) 2 ⌋ + 2. It was proved in [17] that every simple signed K 4 -minor-free graph (G, σ) admits a switching homomorphism to the signed Paley graph SP al 5 , depicted in  We shall prove the following result. Proof. It suffices to show that χ c (F, σ) = 10 3 for the signed outer planar simple graph (F, σ) of Figure 5. Since (F, σ) contains a positive triangle as a subgraph, its circular chromatic number is at least 3. By the formula of the tight cycle the only possible values are 3 and 10 3 . It remains to show that this graph does not admit a circular 3-coloring, that is to say, (F, σ) does not admit a switching homomorphism to K s 6;2 . Note thatK s 6;2 is equivalent to a positive triangle, with each vertex incident to a negative loop. If φ is a switching homomorphism of (F, σ) toK s 6;2 , then at least one negative edge of the negative triangle xyz is mapped to a negative loop, because inK s 6;2 every negative closed walk contains a negative loop. Whichever edge of xyz is mapped to a negative loop, its two end vertices are identified and the resulting signed graph has a negative cycle of length 2. ButK s 6;2 contains no negative even closed walk of length 2, a contradiction. Hence χ c (F, σ) = 10 3 .
In Section 5, we have seen that χ c (SBP) = 4. However, we do not know if there is a signed bipartite planar simple graph reaching the bound 4. Further improvement based on the length of the shortest negative cycle is given in the forthcoming work [16].
Next we consider the circular chromatic number of signed planar simple graphs. Since planar simple graphs are 5-degenerate, by Proposition 6.1, we have χ c (SP) ≤ 6. It was conjectured in [13] that every planar simple graph admits a 0-free 4-coloring. If the conjecture was true, it would have implied the best possible bound of 4 for the circular chromatic number of signed planar simple graphs. However, this conjecture was disproved in [9] using a dual notion. A direct proof of a counterexample is given in [15]. Extending this construction, we build a signed planar simple graph whose circular chromatic number is 4 + 2 3 .
We shall construct a signed planar simple graph Ω with χ c (Ω) = 4 + 2 3 . The construction is broken down into construction of certain gadgets. Similar to the gadget of [9], we start with a mini-gadget depicted in Figure 6 and state its circular coloring property in Lemma 6.5. Note that the minimality of the length implies that the two end points of the interval Lemma 6.5. Assume φ is a circular (4 + α)-coloring of the signed graph (T, π) of Figure 6 with 0 ≤ α < 2.
Proof. Let r = 4 + α and let φ be a circular r-coloring of (T, π). Without loss of generality we may assume that φ(x), φ(y) and φ(z) are on C r in the clockwise order, and assume the interval . This is a contradiction as [φ(z), φ(x)] is longest among the three. As φ(y) is contained in [φ(z), φ(x)], and as y is adjacent to both z and x with a negative edge, we conclude that [φ(z), φ(x)] is of length at least 2. On the other hand, since z and x are adjacent with a negative edge, one of the two intervals, is of length at most r 2 − 1 = 1 + α 2 . As α < 2, the only option is that [φ(x), φ(z)] is of length at most 1 + α 2 . For the other direction, assume ℓ φ;x,y,z < 1 − α 2 , say I φ;x,y,z = [0, β] for some β < 1 − α 2 . Each of a, b, c is joined by a positive edge and a negative edge to vertices in x, y, z. This implies that φ(a), φ(b), φ(c) ∈ As each of the intervals [1, 1 + β + α 2 ] and [3 + α 2 , 3 + α + β] has length strictly smaller than 1, two of the vertices a, b, c are colored by colors of distance less than 1 in C r . But abc is a triangle with three positive edges, a contradiction.
For the "moreover" part, without loss of generality, we assume that t 3 = 0, It is straightforward to verify that φ is a circular r-coloring of (T, π).
By taking α = 2 3 − ǫ and a switching at the vertex z, we have the following formulation of the lemma which we will use frequently. Corollary 6.6. Let (T, π ′ ) be a signed graph obtained from (T, π) by a switching at the vertex z, and let φ be a circular ( 14 3 − ǫ)-coloring of (T, π ′ ) where 0 < ǫ < 2 3 . Then ℓ φ;x,y,z ∈ [ 2 We defineW to be the signed graph obtained from signed Wenger graph of Figure 7 by completing each of the four negative facial triangles to a switching of the mini-gadget of Figure 6. Next we show thatW has a property similar to signed indicators, more precisely: . For any circular r-coloring φ ofW , ℓ φ;u,v ≥ 4 9 .
The proof of Lemma 6.7 is long, and we leave it to the next section. Let Γ be obtained fromW by adding a negative edge uv. Let I = (Γ, u, v). It follows from Lemma 6.7 that for 4 ≤ r < 14 3 , Theorem 6.8. Let Ω = K 4 (I).
Proof. First we show that Ω admits a circular 14 3 -coloring. For r = 14 3 , there is a circular r- 3 , φ(x 5 ) = 4 and φ(z) = φ(t) = 1. We observe that each of the four negative triangles satisfies the conditions of Lemma 6.5, and that the coloring of its vertices can be extended to the inner part of the mini-gadget.
As x 2 is joined to u and w by positive edges, (2) For a depiction of these cases, see Figure 8.
[III] Assume to the contrary (by 1) that φ(z) ∈ [1+η, 4 contradicting the fact that x 3 z is a positive edge.
[IV] Assume to the contrary (by 1) that φ(z) ∈ [ 10 contradicting the fact that x 3 z is a positive edge.
This completes the proof of Claim 7.2. ✸ To complete the proof of Lemma 7.1, we partition the interval ( 5 3 − ǫ, 3 + η) into three parts and consider three cases depending on to which part δ belongs.
As x 2 z is a negative edge, and the distance between the intervals [ 10 3 − ǫ 2 , 11 3 +η −ǫ] and [1, δ −2] is strictly larger than 4 contradicting the fact that x 3 z is a positive edge.
This completes the proof that φ(w) / ∈ ( 5 3 − ǫ, 3 + η). We observe that in this proof, vertex x 1 played no role. In other words, the conclusion holds for the signed subgraph induced on G \ x 1 . In this subgraph a switching at U = {w, x 2 , x 3 , x 4 , x 5 } results in an isomorphic copy where x 4 and x 5 play the role of x 2 and x 3 . Thus for the mapping φ ′ defined as 2 , then by the Lemma 7.1, we have no choice for φ(w). Thus we assume in the rest of the proof that η ≤ 1 − 3ǫ 2 and The two cases will be consider separately.
]. We will update the ranges of φ(x i )'s as depicted in Figure 10. In this figure the range of each φ(x i ) is shown as an interval partitioned to two parts. The full interval represents the restriction we have As ℓ([1+η, 4 3 and zx 3 is a positive edge, the points 1+η, φ(z), φ(x 3 ), 3− 3ǫ 2 occur in C r in this cyclic order. This implies that As occurs in C r in this cyclic order. This implies that By considering the positive edges x 5 t and then x 4 x 5 , similar arguments show that Considering the positive edge x 1 x 2 and the range of φ(x 2 ) given above, a similar argument shows Now consider the negative triangle , contrary to Corollary 6.6. Also φ(u) = 0 cannot be an end point of the interval I φ;x 1 ,x 5 ,u , as 0 is at distance less than 2 3 + ǫ 2 from each of the four end points of the intervals that are the ranges of φ(x 1 ) and φ(x 5 ).
As φ( Figure 11). Since wx 1 and wx 5 are positive edges, we have The proof is similar to the previous case. The positive edge zx 3 and the negative edge tx 4 further restrict the ranges of φ(x 3 ), φ(x 4 ). Then, the new ranges of φ(x 3 ) and φ(x 5 ), together with the positive edges x 3 x 2 and x 4 x 5 further restrict the range of φ(x 2 ), φ(x 5 ). As the computations are very similar to the previous case, we just list the conclusion of this argument: Next we consider the negative triangle vx 3 x 4 . As Similar analysis as in the previous case shows that We will update the ranges of φ(x 2 ), . . . , φ(x 5 ) as depicted in Figure 12.
, contrary to the fact that x 2 z is a negative edge. Thus If φ(x 5 ) ∈ ( 1 3 − ǫ 2 , 4 3 − ǫ 2 ], then d (mod r) (φ(x 5 ), φ(t)) < 1, contrary to the fact that x 5 t is a positive edge. Therefore Considering the positive edge x 1 x 2 and the range of φ(x 2 ) given above, we obtain that Next we consider the negative triangle Figure 15) and since x 5 w, x 1 w are both positive edges, we have that The positive edge zx 3 and the negative edge tx 4 further restrict the ranges of φ(x 3 ) and φ(x 4 ) respectively. Then the new ranges of φ(x 3 ) and φ(x 4 ), through the positive edges x 3 x 2 and x 4 x 5 , further restrict the ranges of φ(x 2 ) and φ(x 5 ). By similar computation as previous cases, we have Next we consider the negative triangle and as x 4 w, x 3 w are both positive edges, we have that We will update the ranges of φ(x i )'s as depicted in Figure 16.
Recall that The intervals I w , I 2 , I 3 , I 4 , I 5 are each of length less than 1, and except for I 3 and I 4 there is no intersection among them. As ℓ(I 4 ) < 1, φ(x 3 ) / ∈ I 4 (since x 3 x 4 is a positive edge). That is again a contradiction because of the 5-cycle wx 2 x 3 x 4 x 5 all whose edges are positive.
This completes the proof of Lemma 6.7.

Questions and Remarks
A notion of a circular coloring of signed graphs was introduced in [8]. It is different from the definition in this paper essentially because the concept of "antipodal" points are defined differently. Both definitions use points on a circle as colors (the discrete version in [8] uses Z k as colors, and we can view elements of Z k as points uniformly distributed on a circle). In [8], a fixed diameter of the circle is chosen, and the antipodal of a point is obtained by flipping the circle along the chosen diameter. Thus for such a coloring, the colors are not symmetric. In particular, for each of the two end points of the chosen diameter, its antipodal is itself. In some sense, the definition in [8] more faithfully extends the coloring of signed graphs that allows 0 (as opposed to 0-free coloring) introduced by Zaslavsky, where 0 is a special color, whose antipodal is 0 itself. We consider the speciality of a certain color to be an undesirable feature. A circular object should be invariant under rotation. In this sense, the circular coloring of signed graphs in this paper more faithfully extends the circular coloring of graphs.
The circular coloring of graphs has been studied extensively in the literature. Many of the results and problems on circular coloring of graphs would be interesting in the framework of signed graphs. We list some specific problems below and believe that there are many more interesting problems.

Jaeger-Zhang conjecture and extensions
For a positive integer k, we have χ c (C −2k ) = 4k 2k−1 . On the other hand, while for a negative odd cycle C −(2k+1) we have χ c (C −(2k+1) ) = 2, for the positive odd cycle C +(2k+1) we have we have χ c (C +(2k+1) ) = χ c (C 2k+1 ) = 2k+1 k . These two facts can be stated uniformly by the following definition. Given ij ∈ Z 2 , we say a closed walk W of a signed graph (G, σ) is of type ij if the number of negative edges of W (counting multiplicity) is congruent to i(mod 2), and the total number of edges (counting multiplicity) is congruent to j(mod 2). For ij ∈ Z 2 we define g ij (G, σ) to be the length of a shortest closed walk of type ij in (G, σ), setting it to be ∞ if there is no such a walk (see [18] for corresponding no-homomorphism lemma and relation to coloring and homomorphism). It is a well-known fact that a homomorphism of a graph onto an odd cycle gives an upper on its circular chromatic number. The following theorem, whose proof we leave the the reader, is an extension of this fact.
Theorem 8.1. Given a positive integer l and a signed graph (G, σ) satisfying The question of mapping planar graphs of odd girth large enough to C 2k+1 was shown by C.Q.

Hadwiger conjecture and extensions
One of the most intriguing conjectures in graph theory is the Hadwiger conjecture which tries to extend the four-color theorem. It claims that any graph without a K k+1 -minor is k-colorable. The case k ≤ 3 of this conjecture is rather easy, but the case k = 4 contains the four-color theorem. As the case k + 1 would imply the case k, the difficulty of the conjecture only increases by k. Catlin [2] introduced a stronger version of the case k = 3 which we restate below using the terminology of signed graphs and notion of circular coloring that we have introduced here. A signed graph (H, π) is said to be a minor of (G, σ) if it is obtained from (G, σ) by a series of the following operations: 1. deleting vertices or edges, 2. contracting positive edge, 3. switching. This conjecture, which is stronger than the Hadwiger conjecture, is known as the Odd-Hadwiger conjecture and using the development in this work can be restated as follows. To generalize this, one may ask: Observe thatK s 2k is the signed graph whose vertices are 1, 2, . . . k where each pair of distinct vertices are adjacent by both a negative edge and a positive edge, and each vertex has a negative loop. It follows from the structure of these signed graphs, in an edge-sign preserving mapping of a signed graph (G, σ) toK s 2k , negative edges introduce no restriction, while vertices connected by a positive edge cannot be mapped to a same vertex. In other words, any such a mapping is a proper k-coloring of the subgraph G + σ induced by the set of positive edges of (G, σ). Recall that a switching homomorphism of (G, σ) tô K s 2k is to find a signature σ ′ equivalent to σ and an edge-sign preserving homomorphism of (G, σ ′ ) tô K s 2k . Therefore, based on the following definition we have the next theorem. We define Theorem 8.6. Given a signed graph (G, σ), we have 2χ + (G, σ) − 2 < χ c (G, σ) ≤ 2χ + (G, σ).
Let f (k) be the answer to Problem 8.5. By Theorem 8.6 one observes that if Conjecture 8.4 holds, then f (k) ≤ 2k. Similarly, considering the result of [7] we have f (k) = O(n √ log n).

Signed planar graphs
Let D be the signed graph on two vertices u and v which are adjacent by two edges: one positive, another negative. This graph normally referred to as digon. It is mentioned that χ c (D) = 4, moreover, given r ≥ 4, if φ is a circular r-coloring of D where φ(u) = 0, then simply by the definition we have φ(v) ∈ (1, r 2 − 1). Thus, by Lemma 5.7, when D is viewed as an indicator, we have χ c (G(D)) = 2χ c (G) where G is a graph (not signed) (this is a restatement of Corollary 4.2). In particular, we have χ c (K 4 (D)) = 8. Noting this is a signed planar mulitgraph and that, by the four-color theorem, every signed planar multigraph without a loop admits an edge-sign preserving homomorphism to it, we obtain χ c (SPM)) = 8 where SPM denotes the class of signed planar multigraphs. Furthermore, we recall that a signed graph with a positive loop admits no circular coloring and that adding a negative loop to a vertex of a signed graphs does not affect its circular chromatic number.
For the class of signed planar simple graphs, the upper bound of 6 follows from the fact that these graphs are 5-degenerate. With our definition of circular chromatic number and development in this work, one may restate a conjecture of [13] as to "circular chromatic number of the class of signed planar simple graphs is 4". However, this conjecture is recently disproved in [9]. The first counterexample provided in [9] is essentially the subgraph K 3 (I) of the signed graph of Theorem 6.8 (they become a same signed graph after a switching). The work of [9] is based on the dual interpretation of the circular four-coloring of signed planar graphs. The examples build there then are based on non-hamiltonian cubic bridgeless planar graphs. The underlying graph of the signed graph of Figure 7 is the dual of Tutte fragment used to build the first example of a non-hamiltonian cubic bridgeless planar graph and referred to as Wenger graph in some literature. This graph itself is used as a building block in a number of coloring results. Noting that a connection to a list coloring problem and circular 4-coloring (of signed planar simple) graphs was established by the 3rd author, [30], we refer to [10] for recent use of this gadget in refuting a similar conjecture.
We note, furthermore, that since in Theorem 6.8 we give the exact value of the circular chromatic number of K 4 (I), one does not expect to improve the lower bound using this particular gadget.
It remains an open problem to decide the exact value of the circular chromatic number of the class of signed planar simple graphs or to improve the bounds (of 14 3 and 6) from either direction.

Girth and planarity
Some of the questions mentioned above can be generalized in the following way: Given an integer l and a class C of signed graphs, such as signed planar graphs or signed K 4 -minorfree graphs, what is the circular chromatic number of signed graphs in C whose underlying graphs have girth l?
As an example, a result of [3] implies that every signed planar graph of girth at least 10 admits a switching homomorphism to the signed graph (K 4 , e) which is the signed graph on K 4 with one negative edge. As this signed graph has circular chromatic number 3, we conclude that Theorem 8.7. For the class SP g≥10 of signed planar graphsG of girth at least 10, we have χ c (SP g≥10 ) ≤ 3.
We do not know if this bound is tight.
In a more refined version of the question one might be given three values of l 01 , l 10 and l 11 and be asked for a best bound on circular chromatic number of signed graphs in C which satisfy g ij (G, σ) ≥ l ij .

Spectrum
In the previous question one may also be asked for the full possible range of circular chromatic number of a given family of signed graphs. For example it is known [6] that a rational number r is the circular chromatic number of a non-trivial K 4 -minor-free graph if and only if r ∈ [2, 8 3 ] ∪ {3}. As for signed K 4 -minor-free simple graphs we extended the upper bound to 10 3 , it remains an open question whether each rational number between 8 3 and 10 3 is the circular chromatic number of a K 4 -minor-free signed simple graph. Spectrum of the circular chromatic number of series-parallel graphs of given girth and circular chromatic number of planar graphs were studied in [14,19,20,26,27]. Similar questions are interesting for signed planar graphs and other families of signed graphs.