On colorings and orientations of signed graphs

A classical theorem independently due to Gallai and Roy states that a graph G has a proper k -coloring if and only if G has an orientation without coherent paths of length k . An analogue of this result for signed graphs is proved in this article.


Introduction
Theorem 1.1 is a classical result characterizing the existence of a proper k-coloring of a graph G in terms of orientations. Theorem 1.1 (Gallai [3] and Roy [8]). For a graph G the following statements are equivalent.
(1) G has a proper k-coloring.
(2) G has an acyclic orientation without coherent paths of length k.
(3) G has an orientation without coherent paths of length k.
The coloring of (G, σ) in [7,11] use color sets in Z. There are other algebraic notions of coloring signed graphs which together with integer coloring all have a common generalization in the concept of permutation coloring by Slilaty [9]. We will give a short survey of these notions and their relationships in Section 2. Arguably, however, integer coloring of signed graphs is the most central of these notions. Evidence enough is the rich theory of chromatic polynomials developed by Zaslavsky [12,14] which broadly generalizes chromatic polynomials of ordinary graphs along with their relationship to matroid theory.
The main result of the present article is Theorem 5.2 which is an analogue of Theorem 1.1 for signed graphs. Theorem 5.1 is also a result of interest.

A survey of different coloring notions
Slilaty's notion of coloring permutation-gain graphs [9] specializes to signed graphs as follows. Let K be a set of colors on which the group {+, −} has a left multiplicative action. A K-coloring of a signed graph (G, σ) is a function f : V (G) → K and the K-coloring is said to be proper when for each edge e with endpoints u and v we have that f (u) = σ(e)f (v). This includes the case for which e is a loop, that is, when u = v. Thus a positive loop prevents any K-coloring from being proper while a negative loop simply requires that the color at its vertex is not fixed under the group action.
Color sets K = M 2t = {−t, . . . , −1, 1, . . . , t} and K = M 2t+1 = {−t, . . . , −1, 0, 1, . . . , t} within Z were already discussed in the introduction. Evidently, K-coloring for K = M 2t or M 2t+1 falls under the notion of permutation coloring. Using color set K being a finite additive group is also possible, but has received less attention. Let's call this type of coloring modular coloring. Again, modular coloring is clearly generalized by permutation coloring. Kang and Steffen have recently made investigations [4,5] into modular coloring. Kim and Ozeki [6] noted that integer coloring and modular coloring are both generalized by Dvořák and Postle's correspondence coloring (a.k.a. DP-coloring) [2]. Permutation coloring is also generalized by correspondence coloring; however, permutation coloring does provides more structure. In particular, permutation coloring has a well-defined notion of chromatic polynomials and extensions of colorings to covering graphs.
Say that the group {+, −} acts on the set K. Hence for each x ∈ K either x is fixed (i.e., x = −x) or {x, −x} is an orbit of size two. Propositions 2.1-2.3 contain some basic observations concerning K-coloring of signed graphs. Proposition 2.1. Say that {+, −} acts on finite sets K 1 and K 2 in which K i has s i fixed elements and t i orbits of order two. If (G, σ) has a proper K 1 -coloring, |K 2 | ≥ |K 1 |, and t 2 ≥ t 1 , then (G, σ) has a proper K 2 -coloring.
Proof. Because |K 2 | ≥ |K 1 | and t 2 ≥ t 1 there is an injective function ι : K 1 → K 2 such that for every x ∈ K 1 in an orbit of size two, −ι(x) = ι(−x). Now if f is a proper K 1 -coloring of (G, σ), then ιf is a proper K 2 -coloring of (G, σ).

Proposition 2.2. Let
A be any additive group with |A| = 2k + 1. If (G, σ) is a signed graph, then there is a bijection between the proper A-colorings of (G, σ) and the proper (2k + 1)-colorings of (G, σ).

Background material
In a graph G, an incidence is where an end of an edge meets a vertex. As such every edge (including loops) has two distinct incidences. An incidence can be denoted by a pair (v, e) in which vertex v is an endpoint of edge e. Although this notation does not distinguish between the two distinct incidences of a loop, it can be modified as (v, e) 1 and (v, e) 2 in order to distinguish the two. A bidirection on G is a function β : I(G) → {1, −1}. When β(v, e) = +1 we think of the incidence as having an arrow pointing at v and when β(v, e) = −1 the incidence has its arrow pointing away from v. Thus bidirections produce three types of edges: introverted, extroverted, and directed (see Figure 1).

Extroverted
Introverted Directed Figure 1: The three types of edges in a bidirected graph.
Again, a signed graph is a pair (G, σ) in which G is a graph and σ : E(G) → {+, −}. A path or cycle C in G is called positive (or negative) when the product of signs on its edges is positive (or negative). A circuit in (G, σ) is a subgraph which is either a positive cycle, two negative cycles which intersect in a single vertex (called a tight handcuff ), or two vertex-disjoint negative cycles along with a minimal connecting path (called a loose handcuff ).
An orientation of a signed graph (G, σ) is a bidirection β satisfying β(v, e)β(u, e) = −σ(e). When e is a loop, this means β((v, e) 1 )β((u, e) 2 ) = −σ(e). As such, each negative edge is either introverted or extroverted and each positive edge has one of two possible directions. An oriented signed graph is a triple Let (G, σ, β) be an oriented signed graph. A path P in (G, σ, β) is coherent when every internal vertex of (P, σ, β) is non-singular. A cycle C in (G, σ, β) is coherent when every vertex of (C, σ, β) is non-singular. It is worth noting that if a cycle C is negative, then (C, σ, β) has an odd number of singular vertices and so is never coherent. A circuit C in (G, σ) is coherent when every vertex in (C, σ, β) is nonsingular. The reader can check that there are exactly two possibilities for a coherent β on a circuit C and if β is one of them, then −β is the other. An orientation β of (G, σ) is acyclic when it contains no coherent circuit. Zaslavsky [13,Corollary 5.3] proved that if (G, σ, β) is acyclic, then (G, σ, β) has a singular vertex.
Given a signed graph (G, σ), a switching function is a function η : V (G) → {+, −}. Define σ η by σ η (e) = η(u)σ(e)η(v) in which u and v are the endpoints of e. (This includes the case for a loop.) If β is a bidirection of G, then ηβ is a bidirection on G. One can think of ηβ as being obtained from β by reversing the arrows at v when η(v) = − and leaving the arrows at v the same when η(v) = +. Since a vertex is singular in (G, σ, β) if and only if it is singular in (G, σ η , ηβ), a signed-graphic circuit C is coherent in (G, σ, β) if and only if it is coherent in (G, σ η , ηβ).
Now if κ is a proper integer coloring of (G, σ), then there is a natural orientation of (G, σ) induced by κ, call it β κ . For each edge e with ends (u, e) and (v, e) there is exactly one choice for orientation β κ (e) so that β κ (u, e)κ(u) + β κ (v, e)κ(v) > 0. Call β κ the orientation induced by κ. Zaslavsky [11] notes that β κ is acyclic when κ is proper. He then uses induced orientations to generalize the work on pairings of acyclic orientations and proper colorings of ordinary graphs by Stanley [10]. If η is a switching function for (G, σ) and κ a proper integer coloring, then ηκ is a proper coloring of (G, σ η ). In fact, κ → ηκ is a bijection between the collection of all proper colorings of (G, σ) and those of (G, σ η ). If κ is a proper integer coloring of (G, σ), then β ηκ = ηβ κ .

Normalizing colorings and acyclic orientations
If β is an acyclic orientation of (G, σ), then there is partitioning L 0 , L 1 , . . . of V (G) which we shall call the canonical level decomposition of (G, σ, β) which is defined as follows. Let L + 0 denote the set of sinks and isolated vertices in (G, σ, β) and let L − 0 denote the set of sources in (G, σ, β).
and L − i+1 be respectively the set of sinks along with isolated vertices and the set of sources in acyclic (G i+1 , σ, β) and let L i+1 = L + i+1 ∪ L − i+1 ; otherwise, we halt the process. The normalization of β is ηβ in which η is a switching function for which η(v) = when v ∈ L i . Note that the canonical level decompositions of β and ηβ are both L 0 , L 1 , . . .. (1) If e is a negative edge, then e is extroverted.
(2) If e is a positive edge with head end in L i and tail end in L j , then i < j.
(3) If v ∈ L j for j > 0, then there is a positive edge e with v as its tail with w ∈ L j−1 as its head.
Proof. (1) If e is an introverted negative edge, then the endpoint(s) of e would be indicated as sources in the construction of the canonical level decomposition. However, since β is normalized, every vertex is indicated as a sink or isolated vertex in the construction of the canonical level decomposition, a contradiction.
(2) If e is a positive edge for which j ≤ i, then its tail would be indicated as a source during the construction of the canonical level decomposition. This yields the same contradiction as in the proof of (1).
Proof. (1 and 2) Since ηκ is normalized, each v ∈ V (G) satisfies ηκ(v) ∈ {1, . . . , t} when n = 2t and ηκ(v) ∈ {0, 1, . . . , t − 1} when n = 2t − 1. We proceed by induction with base case covering L 0 . In the base case, if ηκ(v) has the maximum value, then β ηκ must be a sink or isolated vertex and so v ∈ L + 0 . Conversely if v ∈ L + 0 but ηκ(v) does not have the maximum value, then v cannot have a positive link to any vertex having the maximum value under ηκ. Thus ηκ(v) may be changed to the maximum value without affecting the propriety of the coloring, a contradiction of the maximality of v∈V (G) |κ(v)|. Thus L + 0 = {v : κ(v) = k} when n is even and L + 0 = {v : κ(v) = k − 1} when n is odd. Finally we must show that L − 0 = ∅. By way of contradiction, assume that v ∈ L − 0 . Thus v is not incident to any negative edges; furthermore, every positive neighbor of v must have ηκ-value strictly greater than ηκ(v) and v is not isolated as isolated vertices are placed in L + 0 . Thus without affecting the propriety of ηκ we can change the color ηκ(v) to ηκ(v) = −k when n is even or ηκ(v) = −(k − 1) when n is odd and then renormalize. This, however, contradicts the maximality of v∈V (G) |κ(v)|. Thus L − 0 = ∅, as required. Assume inductively that L i = {v : κ(v) = k − i} when n is even and L i = {v : κ(v) = k − 1 − i} when n is odd for every i ∈ {0, . . . , t}. Thus the maximum ηκ-value among vertices in G t+1 = G − (L 0 ∪ · · · ∪ L t ) is k − (t + 1) when n is even and k−1−(t+1) when n is odd. Thus the argument for the base case applies to (G t+1 , σ, β ηκ ) making L t+1 = {v : κ(v) = k−(t+1)} when n is even and L t+1 = {v : κ(v) = k − 1 − (t + 1)} when n is odd.
(3) Follows from Parts (1) and (2), the maximality of v∈V (G) |κ(v)|, and the fact that any vertex whose color is a maximum must be a sink. (1) (G, σ) has a proper 2k-coloring if and only if (G, σ) has an acyclic orientation whose canonical level decomposition has at most k levels.

Main result
(2) (G, σ) has a proper (2k −1)-coloring if and only if (G, σ) has an acyclic orientation whose canonical level decomposition has at most k levels where level L k−1 is independent or empty.
A balloon in a signed graph is either a negative cycle or a negative cycle along with a path intersecting the cycle at exactly one of its endpoints. The length of a balloon is the length of the cycle plus twice the length of the path. A balloon is coherently oriented when is has exactly one singular vertex. When the balloon includes a path, this singular vertex must therefore be the endpoint of the path that is not on the cycle. When the balloon does not contain a path, then this singular vertex is just some vertex of the cycle. Proof. For both directions of the proof, let k be the unique integer satisfying n ∈ {2k, 2k + 1}; that is, n 2 = k + 1 when n is odd and n 2 = k when n is even. (−→) Let κ be a proper n-coloring of (G, σ) chosen among all proper n-colorings so that v∈V (G) |κ(v)| is a maximum. Let ηκ be the normalization of κ. Proposition 4.2 implies that β ηκ is a normalized acyclic orientation. Since every negative edge in (G, σ, β ηκ ) is extroverted, a coherent path in (G, σ, β ηκ ) cannot contain more than one negative edge. Thus a positive coherent path in (G, σ, β ηκ ) cannot have length n 2 because the colors of the vertices along the path are strictly increasing. A negative coherent path contains exactly one negative edge e and so is of the form P 1 eP 2 in which e is extroverted and P i is a positive coherent path where the colors of the vertices along P i are strictly increasing. When n = 2k, this implies that P 1 eP 2 has length at most 1+2(k −1) = 2k −1, as required. When n = 2k +1, the length of P 1 eP 2 is at most 1+k +(k −1) = 2k because the endpoints of e cannot both have color 0. A coherent balloon has exactly one negative edge, call it e, which is in the cycle of the balloon and then has the form P 1 eP 2 in which P 1 and P 2 are coherent positive paths from the two endpoints of e and which both end at the same vertex. Again, then length of the balloon is at most 2k − 1 when n = 2k and at most 2k when n = 2k + 1, as required.
(←−) Let β be a normalized a acyclic orientation of (G, σ) which contains no positive coherent path of length n 2 , no negative coherent path of length n, and no coherent balloon of length n.
If n = 2k is even, then let the levels for β be called L 1 , . . . , L m . Proposition 4.1 and the fact that there is no coherent positive path of length k = n 2 imply that m ≤ k. Define a 2k-coloring κ by κ(v) = k + 1 − i for v ∈ L i . Proposition 4.1 now implies that κ is a proper coloring.
If n = 2k + 1, then let the levels for β be called L 0 , . . . , L m . Proposition 4.1 and the fact that there is no coherent positive path of length k + 1 = n 2 imply that m ≤ k. Define a (2k + 1)-coloring κ by κ(v) = k − i for v ∈ L i . Proposition 4.1 now implies that κ is a proper coloring excepting the possibility that there is a negative link or loop on vertices of color 0. If such a negative edge e exists, however, then Proposition 4.1 could be used to construct a coherently bidirected balloon or negative path of length n = 2k + 1, a contradiction.
We considered the question of whether or not any two of the three necessary conditions (in particular, conditions (2) and (3)) from Theorem 5.2 might themselves be sufficient to imply the existence of a proper n-coloring. We were not able to find a proof or counterexample.