Symmetric Set Coloring of Signed Graphs

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 underlying graph. These results are used in the second part where we introduce the symset-chromatic number $\chi_{sym}(G,\sigma)$ of a signed graph $(G,\sigma)$. We show that the symset-chromatic number gives the minimum partition of a signed graph into independent sets and non-bipartite antibalanced subgraphs. In particular, $\chi_{sym}(G,\sigma) \leq \chi(G)$. In the final section we show that these colorings can also be formalized as $DP$-colorings.


Introduction
A signed graph (G, σ) is a multigraph G together with a function σ : E(G) → {±}, where {±} is seen as a multiplicative group. The function σ is called a signature of G and σ(e) is called the sign of e. An edge e is negative if σ(e) = − and it is positive otherwise. The set of negative edges is denoted by E − σ , and E(G) − E − σ is the set of positive edges. A multigraph G is sometimes called the underlying graph of the signed graph (G, σ).
Let (G , σ | E(G ) ) be a subgraph of (G, σ). The sign of (G , σ | E(G ) ) is the product of the signs of its edges. A circuit is a connected 2-regular graph. It is positive if its sign is + and negative otherwise. A subgraph (G , σ | E(G ) ) is balanced if all circuits in (G , σ | E(G ) ) are positive, otherwise it is unbalanced. Furthermore, negative (positive) circuits are also often called unbalanced (balanced) circuits. If σ(e) = + for all e ∈ E(G), then σ is the all-positive signature and it is denoted by +, and if σ(e) = − for all e ∈ E(G), then σ is the all-negative signature and it is denoted by -.
A switching of a signed graph (G, σ) at a set of vertices X defines a signed graph (G, σ ) which is obtained from (G, σ) by reversing the sign of each edge of the edge cut ∂ G (X), where ∂ G (X) denotes the set of edges having exactly one end in X, i.e. σ (e) = −σ(e) if e ∈ ∂ G (X) and σ (e) = σ(e) otherwise.
If X = {v}, then we also say that (G, σ ) is obtained from (G, σ) by switching at v.
Switching defines an equivalence relation on the set of all signed graphs on G. We say that (G, σ 1 ) and (G, σ 2 ) are equivalent if they can be obtained from each other by a switching at a vertex set X. We also say that σ 1 and σ 2 are equivalent signatures of G. From Harary's [4] characterization of balanced signed graphs it follows that a signed graph (G, σ) is balanced if and only if it is equivalent to (G, +).
A signed graph (G, σ) is antibalanced if it is equivalent to (G, -). The signed extension ±H of a graph H is obtained from H by replacing every edge by two edges, one positive and one negative.

Motivation
We consider colorings of signed graphs which are defined by assigning colors to its vertices.
Let (G, σ) be a signed graph and S be a set of colors. A function c : V (G) −→ S is a coloring of (G, σ). A coloring c is proper if c(v) = σ(e)c(w) for each edge e = vw. If (G, σ) admits a proper coloring with elements from S, we say that (G, σ) is S-colorable.
While the coloring-condition for positive edges remains unchanged with respect to the unsigned case, the condition on a negative edge e = vw requires that c(v) = −c(w). It implies that −s ∈ S for each s ∈ S. At this point, the choice of the elements of S has strong consequences on the colorings. Indeed, two cases have to be distinguished: when s is a non-self-inverse element, that is s = −s, and when s is a self-inverse element, and so s = −s.
The objective of this paper is to define a coloring and a corresponding chromatic number for a signed graph (G, σ) which gives a minimum color coded partition of the vertex set and which does not depend on the number of self-inverse colors which are admitted for coloring.
To achieve this we will first discuss colorings where the number of self-inverse elements is fixed. This somewhat technical part is performed in Sect. 2, where we determine the chromatic spectrum of signed graphs and prove a Brooks'-type theorem for these kinds of coloring. These results are used in Sect. 3, where we show that the symset chromatic number (which will be defined later in this section) describes the minimum partition of the signed graph into independent sets and antibalanced non-bipartite subgraphs. It follows that this parameter gives a lower bound on the number of pairwise vertex-disjoint negative circuits of a signed graph. We further give an upper bound for the symset-chromatic number for a specific class of signed graphs. In the concluding section we show that circular coloring of signed graphs [7,10] is also covered by our approach and that all these colorings can also be formalized as DP -coloring.

Colorings and Basic Results
If a vertex v of (G, Σ) is incident to a positive loop, then (G, σ) does not have a proper coloring. If v is incident to a negative loop, then it has to be colored with a non-self-inverse color, which somehow counteracts our aforementioned objective. For these reasons, we will consider multigraphs without loops.
Next, we will introduce coloring of signed graphs which covers all approaches of signed graph coloring which are defined by assigning colors to the vertices of the graph. The sets of colors will be symmetric sets.  An element s of a symmetric set S is self-inverse if s = −s. A symmetric set with selfinverse elements 0 1 , . . . , 0 t and non-self-inverse elements ±s 1 , . . . , ±s k is denoted by S t 2k . Clearly, |S t 2k | = t + 2k. A further natural requirement on signed graph coloring is that equivalent signed graphs should have the same coloring properties. Let (G, σ ) be obtained from (G, σ) by switching at a vertex u. If (G, σ) admits a proper coloring c with elements from S t 2k , then c with c (u) = −c(u) and c (v) = c(v) for v = u is a proper coloring of (G, σ ) with elements from Proposition 1.2. Let (G, σ) and (G, σ ) be equivalent signed graphs. Then (G, σ) admits a proper S t 2k -coloring if and only if (G, σ ) admits a proper S t 2k -coloring.
Schweser and Stiebitz [11] used the term symmetric set for subsets Z ⊆ Z with the In case of finite sets this gives symmetric sets with t self-inverse elements for t ∈ {0, 1}. Examples for symmetric sets with more than one self-inverse element are subsets Z of Z k 2n , with Z = −Z . Here the vectors whose entries are either 0 or n are self-inverse.
Self-inverse elements someway annul the effect of the sign, so the following proposition naturally holds. Proposition 1.3. Every signed graph (G, σ) has a proper S χ(G) 0 -coloring, where χ(G) denotes the chromatic number of the underlying graph G.
The main problem in coloring with symmetric sets is that their cardinality and the number self-inverse elements have the same parity. This has some surprising consequences as it can be that the set of colors has more elements than the vertex set of the graph. This issue has been addressed in several ways (see e.g. [13]).
Kang and Steffen [7] introduced cyclic coloring of signed graphs and they used cyclic groups Z n as the set of colors. The cyclic chromatic number, denoted by χ mod (G, σ), is the smallest integer n such that (G, σ) admits a proper coloring with elements of Z n .
Interestingly, these approaches may not only provide different chromatic numbers for the same signed graphs, but they even have different general bounds. On one side an antibalanced triangle is colorable with M 2 by assigning 1 to all its vertices, but it is not Z 2 -colorable. On the other side, the signed extension of the complete graph on 4 vertices, ±K 4 , has a Z 6 -coloring but no M 6 -coloring. Note that here, in both types of coloring the set of colors contains more elements than the vertex set of the graph. The reason for this is given by the different number of self-inverse elements allowed: indeed, in the coloring defined by Zaslavsky we can have either 0 or 1 self-inverse element, while cyclic coloring uses 1 or 2 self-inverse elements.
The Symset-Chromatic Number for all t ≥ χ(G), it follows by Proposition 1.3 that every signed graph (G, σ) has a proper S t 2k -coloring for all t ≥ χ(G). For this reason, we assume t ≤ χ(G) in the following.
If an antibalanced subgraph of (G, σ) which is induced by a non-self-inverse color is bipartite, then the non-self-inverse color can be replaced by two self-inverse colors. In that case, (G, σ) has an S t 2k -and an S t+2 The above examples show that N is not necessarily associated with a unique symmetric set S for which (G, σ) admits a minimum proper S-coloring. To overcome this problem we define max t min{χ t sym (G, σ) : 0 ≤ t ≤ χ(G)} to be the symset chromatic number of (G, σ), which is denoted by χ sym (G, σ). Furthermore, we say that an S t 2k -coloring is minimum if χ sym (G, σ) = t + 2k. By Proposition 1.2 it follows that equivalent signed graphs have the same symset chromatic number.
In preparation for the study of the symset-chromatic number in Sect. 3 we study the symset-t-chromatic number in the next section.

The Symset t-Chromatic Number
An S t 2k -coloring of (G, σ) provides some information on the structure of (G, σ).
Proposition 2.1. If a signed graph (G, σ) admits a proper S t 2k -coloring c, then c induces a partition of V (G) such that c −1 (0) is an independent set in G for every self-inverse color 0, is an antibalanced subgraph of (G, σ) for every non-self-inverse color ±s.
We first prove upper bounds for the symset t-chromatic number in terms of the chromatic number of the underlying graph. The cases t ≤ 1 and t = 2 had been proved in [9] and [5], respectively.
Then, at least k − t vertices of H 1 are colored with pairwise different colors from {±s t+1 · · · ± s l }. Furthermore, for each i ∈ {2, . . . , k − t} each all-negative copy H i of K k contains at least two vertices of the same color of {±s t+1 · · · ± s l }. Since for all 2 ≤ i < j ≤ k − t + 1 each vertex of H i is connected by a positive edge to every vertex of H j , it follows that for all i = j, the multiple used colors in H i are different from the multiple used colors in H j . Thus, at least 2(k − t) pairwise different non-self-inverse colors are needed; k − t for the all-negative copies of K k and k − t for the coloring of H 1 . But we have only 2(l − t) non-self-inverse colors and l < k, a contradiction. Thus,

Brooks' Type Theorem for the Symset t-Chromatic Number
We are going to prove a Brooks' type theorem for the symset t-chromatic number, which implies Brooks' Theorem for unsigned graphs.
The symset t-chromatic number has the same parity as t. Observe that, if t = χ(G) − l, then Theorem 2.2 can be reformulated as χ t sym (G, σ) ≤ χ(G) + l. By parity we obtain equality in the following statement.
If G is a graph with χ(G) = ∆(G) = t + 1, then χ t sym (G, σ) = ∆(G) + 1 by Proposition 2.3. The following Brooks' type statement is the main result of this section.
is an unbalanced even circuit and t = 0 or is an unbalanced odd circuit and t = 2.
We will prove the statement by formulating some propositions, some of which might be of own interest.
Proof. Since we assume that t ≤ χ(C n ), it follows that t ∈ {0, 1, 2, 3}, where t = 3 only applies if n is odd. In this case, we have χ 3 sym (C n , σ) = χ(C n ) = 3. Furthermore, χ 1 sym (C n , σ) = 3 is easy to check. The statements for t = 2 follow with Proposition 2.3. It is easy to see that χ 0 sym (C n , σ) ≤ 4 and χ 0 sym (C n , σ) = 2 if and only if n is even and C n is balanced or n is odd and C n is unbalanced.
The following statement is a standard lemma for coloring.
Lemma 2.7. The vertices of a connected graph G can be ordered in a sequence x 1 , x 2 , ..., x n so that x n is any preassigned vertex of G and for each i < n the vertex x i has a neighbor among x i+1 , ..., x n . Lemma 2.8. Let (G, σ) be a simple connected signed graph. If G is not regular, then Proof. Let v be a vertex having degree d G (v) ≤ ∆ − 1. By Lemma 2.7, there exists an ordering of the vertices x 1 , ..., x n such that x n = v and for each i < n the vertex x i has neighbors among x i+1 , ..., x n . We follow this order to color the vertices by using the greedy algorithm. We can first use the t self-inverse colors, and then add pairs of non-self-inverse colors when it is necessary.
If ∆ − t = 2n is even, then we use exactly n non-self-inverse colors ±s. Each vertex x i , i < n, has at most ∆ − 1 neighbors which have been colored previously. Since it also holds d(x n ) ≤ ∆ − 1, the graph has an S t 2k -coloring, with t + 2k = ∆. If ∆ − t is odd, then the result follows similarly.
For the proof of Theorem 2.4 we also use the following lemma. Then G contains a pair of vertices a and b at distance 2 such that the graph G − {a, b} is connected.

Proof of Theorem 2.4
Proof. Propositions 2.5 and 2.6 imply that the statement is true for complete graphs and circuits. By Lemma 2.8 it suffices to prove it for non-complete regular graphs with a maximum vertex of degree at least 3. We can also assume that the graph is connected.
Let (G, σ) be a signed graph of order n and 0 ≤ t ≤ χ(G). If ∆(G) − t is odd, then (G, σ) can be colored greedily with ∆(G) + 1 colors. Hence, we focus on the case where ∆(G) − t is even. We show that the graph has an S t 2k -coloring with t + 2k ≤ ∆(G). Assume that (G, σ) is 2-connected. By Lemma 2.9 there are two non-adjacent vertices a Thus, it can be colored by ∆(G) colors by Lemma 2.8. By relabeling we can always suppose that v is colored with the same element in each graph, so the entire graph is also S t 2k -colorable, with t + 2k = ∆(G).
As a simple consequence of Theorem 2.2 and Corollary 2.10 we obtain the following statement on the signed extension of a graph. Corollary 2.11. Let G be a connected graph. If G is a complete graph or an odd circuit,
Since |S t 2k | has the same parity as t, it follows that the t-chromatic spectrum contains only values of the same parity.
Proposition 2.12. Let G be a graph and t a positive integer. Then m χ t sym (G) = t + 2.
The coloring can be easily extended to (G, σ) by adding at most two colors, so χ t sym (G, σ) ≤ t + 2k − 2, which is a contradiction.
Proof. First, we stepwise remove vertices v such that the removal of v does not decrease the symset t-chromatic number. The remaining subgraph (G , σ | G ) is λ t -critical.
Secondly, we remove another vertex w from G . Lemma 2.13 implies that this graph has t-chromatic number t + 2k − 2. By proceeding as before, we find a critical subgraph with the same t-chromatic number. This process can be iterated until we obtain a λ i t -critical graph, for each i ∈ {1, ..., k}.
Proof. Let (G, σ) be a signature such that χ t sym (G, σ) = M χ t sym (G) = t + 2k. By Theorem 2.14, we know that for each value of λ i t = t + 2i, where i ∈ {1, ..., k}, (G, σ) has a λ i tchromatic subgraph (H, τ ). Our aim is to prove that the signature τ can be extended to a signature τ in G such that χ t sym (G, τ ) = t + 2i. Let c : V (H) → S t 2i the λ i t -coloring of (H, τ ). For each edge uv ∈ E(G) we define τ in the following way: If v ∈ V (H) and u / ∈ V (H) and v is not colored with 1, τ (uv) = +.
obtain a proper S t 2i coloring, so the statement follows.

The Symset Chromatic Number
Next we skip the constraint on the set of colors given by fixing the value of t and we focus on the symset chromatic number. If c is an S t 2k -coloring, then c −1 (0 j ) is also called a selfinverse color class for each j ∈ {1, . . . , t}. Similarly, c −1 (±s i ) is called a non-self-inverse color class for i ∈ {1, . . . , k}. Clearly, a self-inverse color class is an independent set of the graph, while a non-self-inverse color class induces an antibalanced subgraph. If the color classes are induced by a λ t -coloring of (G, σ) and λ t = χ sym (G, σ), then any non-self-inverse color class induces a non-bipartite subgraph of G, see Proposition 2.1.

The Chromatic Spectrum and Structural Implications
The symset chromatic number gives some information on circuits in the underlying graph G and on the frustration index l(G, σ), which is defined as the minimum number of edges which have to be removed from (G, σ) to make the graph balanced [4].
Proof. For k = 0 there is nothing to prove. So assume k ≥ 1. Let c be an S t 2k -coloring of (G, σ) and let S be a non-self-inverse color class. Since t is maximum, it follows that is not bipartite. Thus, it contains an odd and therefore unbalanced circuit. Since this is true for every subgraph induced by a non-self-inverse color class, the statement follows.
Furthermore, the bound regarding the frustration index is sharp: the graph in Fig. 1 has frustration index 2 and it can be easily seen that a minimum coloring requires two non-self-inverse colors.
Next, we will prove that the symset chromatic spectrum is an interval of integers. Proof. We proceed by induction on the order of the graph. If G is the K 1 , then the statement is trivial. Let us remark that it obviously true for bipartite graphs.
Let i ∈ {2, . . . , χ(G ) − 1} and σ i be a signature of G such that χ sym (G , σ i ) = i. Since i < χ(G ) it follows that i = t + 2k and k ≥ 1. Let c be an S t 2k -coloring of (G , σ i ). We also assume, by switching, that c does not use the negative colors. It implies that all the edges connecting vertices in the same non-self-inverse color class are negative.
Extend σ i to a signature σ i of G as follows. Let vw ∈ E(G). If c (w) is self-inverse, then let σ i (vw) = + and σ i (vw) = − for otherwise. Since v is connected to a non-self-inverse color class by negative edges only, it can be colored with the same color. Thus, χ sym (G, σ i ) ≤ i.
Proof. By the coloring c of (G, σ) we have that χ sym (H p,q , σ p,q )) ≤ p + 2q. However, if there would be a better coloring with less colors or one with the same number of colors but more self-inverse colors, then there would be a better coloring for (G, σ), a contradiction.
We conclude with the following statement. On the other side, assume that (G, σ) has a partition into t independent sets and k antibalanced subgraphs with t + 2k minimum. Then there is a partition P with the maximum number of independent sets. Let t be this number. Then P has k = 1 2 (t + 2k − t) non-bipartite antibalanced subgraphs. Thus, (G, σ) has an S t 2k -coloring and therefore, χ sym (G, σ) = t + 2k.

Upper Bounds for the Symset Chromatic Number
By definition, χ sym (G, σ) ≤ χ(G) and, therefore, Brooks' Theorem can easily be extended to the symset chromatic number. Indeed, if χ sym (G, σ) = χ(G), then χ sym (G, σ) ≤ ∆(G) − 1 unless G is complete or an odd circuit. This statement can be further improved if there is a small non-self-inverse color class.
Proof. Among all λ t -colorings of (G, σ) which have a non-self-inverse color class with precisely three vertices choose coloring c with a maximum number of vertices in the union of the self-inverse color classes and then choose the non-self-inverse S 1 , . . . , S k , such that |S 1 | is maximum, according to the choice of S 1 , . . . , S i choose S i+1 such that |S i+1 | is maximum. Since every non-self-inverse color class has at least three vertices we can assume that Clearly, G[T ] is a triangle. Note that by the choice of c every vertex of T is connected to each self-inverse color class by an edge and to each non-self-inverse color class by a positive and a negative edge. It implies that each vertex v ∈ T has d G (v) ≥ t + 2k.
We will show that there is a vertex v ∈ T with d G (v) ≥ t + 3k − 1.
Proof of the claim. Suppose to the contrary that the claim is not true.
We assume that v 1 w 1 is negative and v 1 w 2 is positive.
Suppose that w 2 is not a neighbor of v i , i ∈ {2, 3}. Then w 2 and v i can be colored with one self-inverse color, v j (j = 1, i) can be colored with another self-inverse color and v 1 with color s, since it is connected by a negative edge to its second neighbor in S. Thus, w 2 is also neighbor of v 2 and v 3 . By switching at T we deduce that w 1 , w 2 are both neighbors of Hence, G[T ∪{w 1 , w 2 }] is a complete signed subgraph (H 5 , σ 5 ) of (G, σ). Clearly, all edges within two vertices of T are negative and all edges between two vertices of S are negative.
We will discuss the following distribution of positive and negative edges: v 1 w 2 , v 2 w 2 , v 3 w 1 are positive and all other edges in (H 5 , σ 5 ) are negative. Furthermore, we can assume that v 2 w 3 is negative (see Figure 2). The argumentation for other distributions is similar.
If w 3 = w 4 and v 3 w 3 is negative or w 3 = w 4 , then color v 3 and w 2 with two self-inverse colors and the remaining vertices with color s to obtain the desired contradiction.

Circular Coloring
Circular coloring is a well studied refinement of ordinary coloring of graphs. Here the set of colors is provided with a (circular) metric. Kang and Steffen [7] used elements of cyclic groups as colors for their definition of (k, d)-coloring of a signed graph (G, σ). For positive integers k, d with k ≥ 2d, a (k, d)-coloring of a signed graph (G, σ) is a map c : V (G) → Z k such that for each edge e = vw, |c(v) − σ(e)c(w)| ≥ d mod k. Hence, this coloring is a specific S 1 2k -coloring if k = 2k + 1, and a specific S 2 2(k −1) -coloring if k = 2k . Naserasr, Wang and Zhu [10] generalized circular coloring of graphs to signed graphs as follows. For i, j ∈ {0, 1, . . . , p − 1}, the modulo-p distance between i and j is d ( mod p) (i, j) = min{|i − j|, p − |i − j|}. For an even integer p, the antipodal color of x ∈ {0, 1, . . . , p − 1} is x = x + p 2 mod p. Let p be an even integer and q ≤ p 2 be 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 each positive edge xy, d ( mod p) (f (x), f (y)) ≥ q, and for each negative edge xy, d ( mod p) (f (x), f (y)) ≥ q. Now it is easy to see that this defines a specific S 0 p -coloring of (G, σ).

DP-Coloring
In this subsection, we show that coloring of signed graphs with elements from a symmetric set can be described as special DP -coloring. The DP -coloring was introduced for graphs by Dvořák and Postle [3] under the name correspondence coloring. We follow Bernshteǐn, Kostochka, and Pron [1] and consider multigraphs.
Let G be a multigraph. A cover of G is a pair (L, H), where L is an assignment of pairwise disjoint sets to the vertices of G and H is the graph with vertex set v∈V (G) L(v) satisfying the following conditions: Let t, k be positive integers and L t 2k = {s 1 , . . . , s t , r 0 , . . . , r 2k−1 }. An (L, H t 2k )-cover of a signed multigraph is a cover of (G, σ) with L(v) = L t 2k for each vertex v ∈ V (G) and H t 2k satisfies the following conditions: 1. H t 2k [L(v)] is an independent set for each v ∈ V (G).
2. If there is no edge between u and w, then E H t 2k (L(u), L(w)) = ∅. It is easy to see that a signed graph is S t 2k -colorable if and only if it (L, H t 2k )-colorable. The associated chromatic numbers are to be defined accordingly.
If we consider coloring of signed graphs we can restrict to multigraphs with edge multiplicity at most 2, since more than one positive and one negative edge between two vertices do not have any effect on the coloring properties of the multigraph. That is, if we consider (L, H t 2k )-cover of a signed extension ±G of a graph G, then H t 2k [E H (L(u), L(w)] is a 2-regular multigraph whose components are digons and circuits of lengths 4, for any two adjacent vertices u, v of G.
However, the DP-coloring approach allows further flexibility and generalizations. For instance, DP-coloring is considered in the more general context of gain graphs in a short note of Slilaty [12], where the corresponding chromatic polynomials are defined.

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