A categorification for the signed chromatic polynomial
aa r X i v : . [ m a t h . C O ] J a n A CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL
ZHIYUN CHENG, ZIYI LEI, YITIAN WANG, AND YANGUO ZHANGA
BSTRACT . By coloring a signed graph by signed colors, one obtains the signed chromatic polynomialof the signed graph. For each signed graph we construct graded cohomology groups whose gradedEuler characteristic yields the signed chromatic polynomial of the signed graph. We show that thecohomology groups satisfy a long exact sequence which corresponds to signed deletion-contractionrule. This work is motivated by Helme-Guizon and Rong’s construction of the categorification for thechromatic polynomial of unsigned graphs. keywords signed graph, signed chromatic polynomial, chromatic cohomology1. I
NTRODUCTION
The chromatic polynomial, which encodes the number of distinct ways to color the vertices ofa graph, was introduced by Birkhoff in attempt to attack the four-color problem [1, 2]. Birkhoff’sdefinition is limited to the planar graphs, later Whitney extended this notion to nonplanar graphs.Although this polynomial did not lead to a solution to the four-color problem, it is one of the mostimportant polynomials in graph theory. The reader is referred to [14] for a nice introduction to thechromatic polynomial and [8] for some recent breakthrough.In recent years, there are a lot of works on chromatic polynomial and its categorification. Moti-vated by Khovanov’s seminal work on the categorification of Jones polynomial [9], Helme-Guizonand Rong introduced a categorification for the chromatic polynomial by constructing graded coho-mology groups whose graded Euler characteristic is equal to the chromatic polynomial of the graph[5]. Later, Jasso-Hernandez and Rong introduced a categorification for the Tutte polynomial [7].By using the similar idea, Luse and Rong proposed a categorification for the Penrose polynomial,and put forward some relations with other categorifications [11]. Recently, Sazdanovic and Yip con-structed a categorification of the chromatic symmetric function [15], which can be considered as ageneralization of the chromatic polynomial.The chromatic cohomology was well studied during the past fifteen years. For example, in [3]M. Chmutov, S. Chmutov and Y. Rong proved the knight move theorem for chromatic cohomology.It follows that the ranks of the cohomology groups are completely determined by the chromaticpolynomial. We remark that the original knight move conjecture is false for Khovanov homology[13]. The reader is referred to [6, 10] for some investigations on the torsion in chromatic cohomology.A signed graph is a graph in which each edge is labeled with a positive sign or a negative sign.The signed graph coloring was first studied by Cartwright and Harary in [4]. In early 1980’s, Za-slavsky tried to use signed colors to color signed graphs [17]. The main principle of how to colora signed graph is equivalent signed graphs have the same chromatic number. Here two signedgraphs are said to be equivalent if they are related by finitely many vertex switchings. Zalavskyfound some properties of signed graphs and introduced two kinds of chromatic polynomial, saythe chromatic polynomial and the balanced chromatic polynomial. Recently, a good survey on thistopic was written by Steffen and Vogel [16].It’s natural to ask whether we can define a categorification for the chromatic polynomial and thebalanced chromatic polynomial of signed graphs, following the categorification for the chromaticpolynomial of unsigned graphs. The main aim of this paper is to construct such two categorifi-cations. Furthermore, we can also put forward the so-called signed deletion-contraction rule. We
Mathematics Subject Classification.
Key words and phrases. signed graph, chromatic polynomial, balanced chromatic polynomial, categorification. show that the cohomology groups satisfy a long exact sequence corresponding to it, which is basedon the corresponding exact sequence in Helme-Guizon and Rong’s work [5].The rest of this paper is arranged as follows. Section 2 is devoted to give a brief introductionto the chromatic polynomial. In Section 3, we give a quick review of basics of signed graphs andsigned graph colorings. Then we recall the notion of signed chromatic polynomial, which combinesthe chromatic polynomial and the balanced chromatic polynomial of signed graphs. In the begin-ning of Section 4, we recall the notion of graded dimension of graded Z -modules, then constructthe categorifications for the chromatic polynomial and the balanced chromatic polynomial. Severalexamples are also given. Some basic properties of the cohomology groups in these two categorifica-tions are discussed in Section 5. 2. T HE CHROMATIC POLYNOMIAL
We begin our discussion with a brief introduction to the chromatic polynomial. We shall consis-tently use G to denote a graph, and use V ( G ) , E ( G ) to denote its vertex set and edge set respectively.A proper coloring of G is an assignment of elements from a color set to V ( G ) , such that adjacent ver-tices receive different colors. In other words, a proper coloring of G is a map ϕ from V ( G ) to a colorset C , which requires that for any v , v ∈ V ( G ) , if there exists an edge e ∈ E ( G ) connecting v and v , then ϕ ( v ) = ϕ ( v ) . If the color set C = {
1, 2, · · · , λ } , then we denote the number of all propercolorings of G by P G ( λ ) . It follows immediately that if G contains a loop, i.e. an edge that connectsa vertex to itself, then P G ( λ ) =
0. By using the deletion-contraction relation P G ( λ ) = P G − e ( λ ) − P G / e ( λ ) ,it is not difficult to find that actually P G ( λ ) defines a polynomial [14], which is called the chromaticpolynomial of G . Here G − e denotes the graph obtained from G by removing the edge e , and G / e isthe graph obtained from G by contracting the edge e .It’s obvious that P N k = λ k , if N k is the graph consists of k vertices but zero edges. Together withthe deletion-contraction relation, these two relations uniquely determines P G ( λ ) . As an example,the chromatic polynomial of P = can be calculated as follows P λ ! = P λ ! − P λ ! = P λ ! − P λ ! = P λ ! − P λ ! = ( P λ ! − P λ ! ) − ( P λ ! − P λ ( ))= P λ ! − P λ ! + P λ ( )= λ − λ + λ On the other hand, according to the principle of inclusion-exclusion, there is another formula for P G ( λ ) . For each s ⊆ E ( G ) , denote [ G : s ] the subgraph whose vertex is V ( G ) and edge set is s , andlet k ( s ) be the number of connected components of [ G : s ] . Then we have P G ( λ ) = ∑ s ⊆ E ( G ) ( − ) | s | λ k ( s ) = ∑ i ≥ ( − ) i ∑ s ⊆ E ( G ) , | s | = i λ k ( s ) . CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 3
Note that λ k ( s ) is nothing but the polynomial which counts the ways of colorings of [ G : s ] such thatadjacent vertices have the same color. The formula above plays an important role in the categorifi-cation of the chromatic polynomial, see [5] for more details.3. S IGNED GRAPH AND SIGNED CHROMATIC POLYNOMIALS
Signed graph and signed colorings.
First, let’s take a quick review of signed graphs. Let SG = ( G , σ ) be a signed graph on an ordinary graph G = ( V ( G ) , E ( G )) with a sign on each edge σ : E ( G ) → { − } . For each subgraph of SG , we call it unbalanced if there is a negative circuit, onwhich the product of edges’ signs is negative. Otherwise we say it is balanced . Let us use b ( SG ) todenote the number of balanced components of SG . The following is an example. Example . For the signed graph in Figure 1, there are only 2 negative circuits, say, v v v v and v v v v v . Both of them come from the left component. If follows that the left component is unbal-anced and the right one is balanced, hence we have b ( SG ) = v v v v v v v v + +++ − + −− + F IGURE
1. A signed graph with one balanced component and one unbalanced componentIn [17], Zalavsky introduced the signed coloring on signed graph. Recently, this notion wasmodified by E. M´aˇcajov´a, A. Raspaud and M. ˇSkoviera [12] to make the corresponding chromaticnumber coincides with the classical chromatic number when all the signs of the signed graph arepositive. A signed λ -coloring is map k : V ( SG ) → K , where K = {− µ , · · · , −
1, 0, 1, · · · , µ } if λ = µ +
1, and K = {− µ , · · · , −
1, 1, · · · , µ } (which is called zero-free ), if λ = µ . We call k a proper λ -coloring (or just coloring for short) if for each edge e = v v (it is possible that v = v ), we have k ( v ) = σ ( e ) k ( v ) . The signed chromatic number is defined to be the smallest λ such that SG admits aproper λ -coloring.Let SG = ( G , σ ) be a signed graph, each function u : V ( SG ) → {−
1, 1 } leads to a switching func-tion on SG : for any e ∈ E ( SG ) connecting v , v ∈ V ( SG ) , σ ( e ) will be changed into u ( v ) σ ( e ) u ( v ) .In particular, if the edge e is a loop, then the sign σ ( e ) is preserved. If a switching function sends avertex v to − vertex switching on v and denote thenew signed graph obtained by v ( SG ) . Two signed graphs are called equivalent if they are related byfinitely many vertex switchings. Note that vertex switching preserves the sign of any circuit, hencethe number of balanced components is invariant under vertex switching. On the other hand, it isobvious that if we are given a proper coloring of SG , after applying the vertex switching on somevertex v ∈ V ( SG ) , the coloring obtained from the original coloring by reversing the sign of k ( v ) is aproper coloring for v ( SG ) . It follows that for any given λ , equivalent signed graphs have the samenumber of proper colorings. In particular, if we are given a balanced signed graph, since a balancedsigned graph is equivalent to the corresponding positive graph, it suffices to consider the propercolorings on the unsigned graph.3.2. Signed chromatic polynomials.
Similar to the ordinary graph, the number of proper signedcolorings with λ (odd or even) colors is a polynomial respect to λ , we call it the signed chromaticpolynomial and use P SG ( λ ) to denote it. In particular, if λ = µ + ( µ ∈ Z ) , we call it the chromaticpolynomial , otherwise we call it the balanced chromatic polynomial , as suggested in [17]. ZHIYUN CHENG, ZIYI LEI, YITIAN WANG, AND YANGUO ZHANG
As we mentioned before, a switching function does not change the number of proper colorings,hence it also preserves the signed chromatic polynomial. Similar to the classical chromatic polyno-mial, we have the following deletion-contraction relation.
Proposition 3.1.
Let SG be a signed graph and e a positive edge. We use SG − e to denote the signed graphobtained from SG by deleting e, and use SG / e to denote the signed graph obtained from SG by contracting e.The deletion-contraction relation of the signed graph SG with respect to e readsP SG ( λ ) = P SG − e ( λ ) − P SG / e ( λ ) .Notice that the positive edge e above could be a loop, in which case P SG ( λ ) =
0. If e is a negativeedge but not a loop, we can take vertex switching on one of its endpoint first and then apply thedeletion-contraction relation.Let us use SN nm ( m ≥ n ) to denote the graph with m vertices and n negative loops, where eachnegative loop joins one vertex to itself. The signed chromatic polynomial of SN nm can be calculateddirectly P SN nm ( λ ) = ( λ m − n ( λ − ) n , λ is odd; λ m , λ is even.By using the deletion-contraction relation and the signed chromatic polynomial of SN nm , we cancalculate the signed chromatic polynomial of any signed graph, and verify that both the chromaticpolynomial and the balanced chromatic polynomial are well defined as polynomials. We need topoint out that the signed chromatic polynomial, as a joint name for both, is not always a polynomialstrictly. As an example, let us calculate the signed chromatic polynomial of SP = ++ − . Example . P λ ( SP ) = P λ ++ ! − P λ − + ! = P λ ++ ! − P λ + ! + P λ − ! = λ ( λ − ) − λ ( λ − ) + P λ − ! = ( λ − λ + λ − λ is odd; λ − λ + λ , λ is even.Similar to the unsigned graph, there is another formula for the signed chromatic polynomialdeduced from the principle of inclusion-exclusion. Proposition 3.2.
Let SG be a signed graph on G, we define Q SG ( λ ) to be the polynomial which calculates theways to coloring SG such that for any e ∈ E ( G ) connecting v , v ∈ V ( G ) , it satisfies k ( v ) = σ ( e ) k ( v ) . Then we have Q SG ( λ ) = ( λ b ( SG ) , λ is odd or SG is balanced; else.Proof. For any coloring satisfies the condition above, the coloring of one component is completelydetermined by the color on one vertex. If the component is balanced then there are λ colors tochoose. However, if the component is unbalanced, we can only assign 0 to it. In other words, wecan not color it when λ is even and there is only one way to color it when λ is odd. Then we obtain Q SG ( λ ) = ( λ b ( SG ) , λ is odd ; λ b ( SG ) · n − b ( SG ) , λ is even. CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 5 where n is the number of components of SG and as usual we set 0 =
1. The proof is finished.
Proposition 3.3.
According to the principle of inclusion-exclusion, we haveP SG ( λ ) = ∑ i ≥ ( − ) i ∑ s ⊆ E ( G ) , | s | = i Q [ SG : s ] ( λ ) , where [ SG : s ] is the signed subgraph on [ G : s ] .Proof. We obtain this directly by applying the principle of inclusion-exclusion together with thedefinition of Q SG ( λ ) .4. A CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL
As we mentioned above, the signed chromatic polynomial itself is not always a polynomial.However, the chromatic polynomial and the balanced chromatic polynomial are both well definedas polynomials. So in this section, we give the categorifications for these two polynomials respec-tively. We first recall the definition and some properties of the graded dimension (also called quan-tum dimension) of a graded Z -module.4.1. Graded dimension of graded modules.Definition 4.1.
Let M = L j M j be a graded Z -module, where M j denotes the set of homogeneouselements with degree j . The graded dimension (also called quantum dimension ) of M is the powerseries q dim M : = ∑ j q j · rank (cid:0) M j (cid:1) ,where rank (cid:0) M j (cid:1) = dim Q (cid:0) M j ⊗ Q (cid:1) .We remark that the torsion part of M can not be detected by the graded dimension. Let M and N be two graded Z -modules. Then M ⊕ N and
M ⊗ N are both graded Z -modules and the gradeddimensions can be obtained as below q dim ( M ⊕ N ) = q dim ( M ) + q dim ( N ) , q dim ( M ⊗ N ) = q dim ( M ) · q dim ( N ) .The following example of graded Z -module, which is taken from [5], will be frequently usedthroughout this paper. It was originally used in the construction of Khovanov homology [9]. Example . Let M be the graded free Z -module with two basis elements 1 and x , whose degreesare 0 and 1 respectively. According to the definition of the graded dimension, we have q dim ( M ) = q · rank ( Z ) + q · rank ( Z x ) = + q .On the other hand, by using the left identity above, as M = Z ⊕ Z x , we obtain the same result q dim ( M ) = q dim ( Z ) + q dim ( Z x ) = + q .Additionally, the tensor product formula above tells us that q dim M ⊗ k = ( + q ) k . Definition 4.2.
Let { ℓ } be the “degree shift” operation on graded Z -modules. That is, if M = ⊕ j M j is a graded Z -module where M j denotes the set of elements of M of degree j , we set M{ ℓ } j : = M j − ℓ so that q dim M{ ℓ } = q ℓ · q dim M . In other words, all the degrees are increased by ℓ .For example, since deg x =
1, then Z { } = Z x . And for each ℓ ∈ N , it’s easy to check that M ⊗ Z { ℓ } ∼ = M{ ℓ } , the Z -module isomorphic to M with degree of every homogeneous elementraised up by ℓ .4.2. A categorification for the chromatic polynomial.
In this section, we give a categorification forthe chromatic polynomial, i.e. the signed chromatic polynomial with odd λ . In order to do this, weneed to introduce a sequence of graded Z -modules and graded differentials. ZHIYUN CHENG, ZIYI LEI, YITIAN WANG, AND YANGUO ZHANG
Cochain groups on signed graphs.
Let SG be a signed graph on G , for each signed spanninggraph [ SG : s ] led by s ⊆ E ( SG ) , we assign a graded Z -module M s ( SG ) as follows: we assign acopy of M to each balanced component and a copy of Z to the unbalanced component and then takethe tensor product. In this way, it is guaranteed that q dim M s ( SG ) = Q [ G : s ] ( + q ) if q = λ − ∈ N ,which further ensures the following result. Proposition 4.3.
Let SG be a signed graph. For each signed spanning graph [ SG : s ] led by s ⊆ E ( SG ) ,we define the cochain group C i ( SG ) = L s ⊆ E ( SG ) , | s | = i M s ( SG ) , then P SG ( + q ) = ∑ i ≥ ( − ) i qdimC i ( SG ) ,provided that q ∈ N . Now we have a cochain group C • ( SG ) whose graded Euler characteristic is equal to the chromaticpolynomial with λ = + q , where λ is odd and q is even. The next step is to introduce a differential d s for the chain complex which satisfies d s = The differential.
We first recall the definition of enhanced state of an unsigned graph [5]. Let G = ( V ( G ) , E ( G )) be a graph with an ordering on E ( G ) . An enhanced state S = { s , c } consists of asubset s ⊂ E ( G ) and an assignment c which assigns 1 or x to each component of [ G : s ] . For eachenhanced state S , we set i ( S ) = | s | and j ( S ) to be the number of x in c .We define a multiplication m : M ⊗ M → M by m ( ⊗ ) = m ( ⊗ x ) = m ( x ⊗ ) = x and m ( x ⊗ x ) =
0. We remark that actually there is a Frobenius algebra structure on M [9]. However,here the only operation we need is the multiplication, since adding an edge never increases thecomponent number.We set C i , j ( G ) = span h S | S is an enhanced state of G with i ( S ) = i , j ( S ) = j i , where the span istaken over Z . The differential d : C i , j ( G ) → C i + j ( G ) is defined as d : S = ( s , c ) → ∑ e ∈ E ( G ) − s ( − ) n ( e ) S e ,where n ( e ) is the number of edges in s that are ordered before e . Here S e denotes either an enhancedstate or 0, which is defined as follows • if e connects a component E i to itself, then the components of [ G : s ∪ { e } ] are E , · · · , E i ∪{ e } , · · · , E k ( s ) . We define c e ( E ) = c ( E ) , · · · , c e ( E i ∪ { e } ) = c ( E i ) , · · · , c e ( E k ( s ) ) = c ( E k ( s ) ) . • If e connects two components E i and E j ( i < j ) , then the components of [ G : s ∪ { e } ] are E , · · · , E i − , E i ∪ E j ∪ { e } , E i + , · · · , E j − , E j + , · · · , E k ( s ) . We define c e ( E i ∪ E j ∪ { e } ) = m ( c ( E i ) ⊗ c ( E j )) and c e ( E l ) = c ( E l ) if l = i , j .In particular, if c ( E i ) = c ( E j ) = x , then c e ( E i ∪ E j ∪ { e } ) = m ( x ⊗ x ) =
0. In this case, we set S e = c e is a coloring and we define S e to be the enhanced state ( s e = s ∪ { e } , c e ) . It was provenin [5] that d =
0. Therefore C • ( G ) = { L j ≥ C i , j ( G ) , d } is a graded cochain complex with gradedEuler characteristic the chromatic polynomial P G ( λ ) evaluated at λ = + q . The correspondingcohomology groups H • ( G ) are called the chromatic cohomology groups of G .Now we turn to the enhanced states for the chromatic polynomial of signed graphs. As before,for a given signed graph SG , we use G to denote the corresponding unsigned graph. Definition 4.4.
An enhanced state S = ( s , c ) of G is an enhanced state of SG with respect to chromaticpolynomial if and only if c assigns 1 for all unbalanced components of [ G : s ] . Now we can define C i , j ( SG ) as follows C i , j ( SG ) = span h S | S is an enhanced state of SG with i ( S ) = i , j ( S ) = j i ,where i ( S ) = | s | and j ( S ) = the number of components that x is assigned to.It follows immediately that C i ( SG ) = L j ≥ C i , j ( SG ) . In order to define the differential d s , we needto define a map f from C i , j ( G ) to C i , j ( SG ) . CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 7
Definition 4.5.
Let SG be a signed graph on G , for each enhanced state S = ( s , c ) in C i , j ( G ) , wedefine f ( S ) = ( S , if c assigns 1 to all unbalanced components0, otherwiseIt’s obvious that f extends to a linear map from C i , j ( G ) onto C i , j ( SG ) , as well as C i ( G ) onto C i ( SG ) . We will use f to denote both of them, if there is no confusion. By using f , we can define thedifferential d s according to the following commutative diagram. C n ( G ) d / / f (cid:15) (cid:15) C n + ( G ) f (cid:15) (cid:15) C n ( SG ) d s / / C n + ( SG ) Proposition 4.6.
The differential d s : C n ( SG ) → C n + ( SG ) , defined by d s = f ◦ d ◦ f − is well defined.Proof. Because both f , d are linear functions, it suffices to prove that for each z ∈ C n ( G ) with f ( z ) =
0, we have f ◦ d ( z ) =
0. Recall that C n ( G ) = L s ⊆ E ( G ) , | s | = n M s ( G ) , where M s ( G ) = M ⊗ k ( s ) , we suppose z = m ∑ i = a i S i ( a i = ) , where S i = ( s i , c i ) ( ≤ i ≤ m ) are distinct enhanced states. Since f ( z ) =
0, itfollows that for each 1 ≤ i ≤ m , there exists at least one unbalanced component of [ G : s i ] which isassigned an x . For S , let us choose an unbalanced component E with c ( E ) = x . We break downinto three cases.(1) If an edge e ∈ E ( G ) − s is not adjacent to E , the the corresponding assignment of ( S ) e alsoassigns an x to E . Hence ( S ) e is also sent to 0 according to the definition of f .(2) If an edge e ∈ E ( G ) − s connects E with another component E , then the new component E ∪ E ∪ { e } in [ G : s ∪ { e } ] is unbalanced. If c ( E ) =
1, then ( c ) e ( E ∪ E ∪ { e } ) = m ( x ⊗ ) = x , hence f (( S ) e ) =
0. If c ( E ) = x , then ( c ) e ( E ∪ E ∪ { e } ) = m ( x ⊗ x ) = f .(3) If an edge e ∈ E ( G ) − s joins E to itself, then the new component E ∪ { e } in [ G : s ∪ { e } ] is also unbalanced and ( c ) e ( E ∪ { e } ) = c ( E ) = x , it follows that f (( S ) e ) = S i , the proof is analogous to that of S . In summary, we have f ◦ d ( z ) =
0, which meansthat d s = f ◦ d ◦ f − is well defined. (cid:3) As d = d s = f ◦ d ◦ f − , we can directly obtain d s = f ◦ d ◦ f − =
0, then we obtain thefollowing chain complex C ( SG ) d s −→ C ( SG ) d s −→ C ( SG ) d s −→ · · · d s −→ C i ( SG ) d s −→ · · · . Definition 4.7.
For a given signed graph SG , we call the cohomology groups of the cochain complexabove the chromatic cohomology groups of SG and use H i ( SG ) to denote the i -th chromatic cohomol-ogy group of SG .Since d s is degree preserving, the chromatic cohomology group H i ( SG ) can be decomposed as L j ≥ H i , j ( SG ) . The chromatic polynomial can be recovered by P SG ( + q ) = ∑ i ≥ ( − ) i q dim C i ( SG ) = ∑ i ≥ ( − ) i q dim H i ( SG ) . The following proposition tells us that chromatic cohomology groups arewell defined signed graph invariants. Proposition 4.8.
The chromatic cohomology groups are independent of the order of the edges.Proof.
The proof follows the outline given in [5]. Suppose E ( SG ) = { e , · · · , e k , e k + , · · · , e n } , itsuffices to prove that H i ( SG ) ∼ = H i ( SG ′ ) , where SG ′ is the same signed graph as SG but the edgesare reordered as { e , · · · , e k + , e k , · · · , e n } . ZHIYUN CHENG, ZIYI LEI, YITIAN WANG, AND YANGUO ZHANG
Since C i ( SG ) = L s ⊆ E ( SG ) , | s | = i M s ( SG ) , where M s ( SG ) = M ⊗ b ([ SG : s ]) , it is enough to define an iso-morphism g restricted on M s ( SG ) . Consider the map g s from M s ( SG ) to M s ( SG ′ ) , which is definedto be g s = ( − id , if { e k , e k + } ⊂ sid , otherwiseNow we define the map g : C i ( SG ) → C i ( SG ′ ) as g = L s ⊆ E ( SG ) , | s | = i g s . It is a routine exercise to checkthat g is a chain map which induces an isomorphism between H i ( SG ) and H i ( SG ′ ) . (cid:3) We end this subsection with a concrete example.
Example . Let us consider the signed graph SP , where E ( SP ) = { e , e , e } , and σ ( e ) = σ ( e ) =+ σ ( e ) = −
1, as Figure 2 shows. v v v e , + e , + e , − F IGURE
2. A signed graph SP For each enhanced state S = ( s , c ) of SP , we denote s and c by elements of {
0, 1 } and { x } ,such that the i -th position of s is 1 (0) if e i ∈ s ( e i / ∈ s ) , and all the vertices of the same componenthave the same color. For example, the enhanced states shown in Figure 3 are ( x ) , ( x x ) and ( ) . 1 1 x + x x + − F IGURE
3. Three enhanced states
CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 9
One computes C ( SP ) = span h ( xxx ) , ( xx ) , ( x x ) , ( xx ) , ( x ) , ( x ) , ( x ) , ( ) i ; C ( SP ) = span h ( xxx ) , ( xxx ) , ( xxx ) , ( x ) , ( xx ) , ( xx ) , ( x ) , ( x x ) , ( x ) , ( ) , ( ) , ( ) i ; C ( SP ) = span h ( xxx ) , ( xxx ) , ( xxx ) , ( ) , ( ) , ( ) i ; C ( SP ) = span h ( ) i . B ( SP ) = B ( SP ) = span h ( xxx ) + ( xxx ) , ( xxx ) + ( xxx ) , ( xxx ) + ( xxx ) , ( x x ) + ( xx ) + ( x ) , ( x ) + ( xx ) + ( xx ) , ( x x ) + ( x ) + ( xx ) , ( ) + ( ) + ( ) i ; B ( SP ) = span h ( xxx ) + ( xxx ) , ( xxx ) + ( xxx )( ) + ( ) , ( ) + ( ) i ; B ( SP ) = span h ( ) i . Z ( SP ) = span h ( xxx ) i ; Z ( SP ) = span h ( xxx ) , ( xxx ) , ( xxx ) , ( x ) − ( xx ) , ( xx ) − ( x ) , ( x x ) − ( x ) , ( x ) + ( x ) + ( x ) , ( ) + ( ) + ( ) i ; Z ( SP ) = span h ( xxx ) , ( xxx ) , ( xxx ) , ( ) + ( ) , ( ) + ( ) i ; Z ( SP ) = span h ( ) i .It follows that H ( SP ) = Z ( SP ) / B ( SP ) ∼ = Z { } ; H ( SP ) = Z ( SP ) / B ( SP ) ∼ = Z { } ⊕ Z { } ; H ( SP ) = Z ( SP ) / B ( SP ) ∼ = Z { } ; H ( SP ) = Z ( SP ) / B ( SP ) ∼ = By using Proposition 4.3, we can calculate the chromatic polynomial P SP ( + q ) = ∑ i ≥ ( − ) i q dim C i ( SP ) = ∑ i ≥ ( − ) i q dim H i ( SP ) = q − q + q = q ,where q is even. On the other hand, this result can also be obtained by using the deletion-contractionrelation on the signed graph, which is shown in Example 3.2.4.2.3. A categorification of the deletion-contraction relation.
The aim of this subsection is to introducea long exact sequence, which recovers the deletion-contraction relation given in Proposition 3.1 ifone takes the Euler characteristic of this long exact sequence. So in some sense, the long exactsequence in Corollary 4.10 can be considered as a categorification of the deletion-contraction rule inProposition 3.1.
Theorem 4.9.
Let SG be a signed graph and e a positive edge, we have the following short exact sequence → C •− ( SG / e ) → C • ( SG ) → C • ( SG − e ) → .Proof. Without loss of generality, let us choose an order of E ( SG ) such that the positive edge e is thefirst edge. The order of E ( SG ) induces the an order for E ( SG / e ) and E ( SG − e ) .We first introduce a morphism of cochain complexes, say e m : C • ( SG − e ) → C • ( SG / e ) . Recallthat C i ( SG ) = L s ⊆ E ( SG ) , | s | = i M s ( SG ) , where M s ( SG ) = M ⊗ b ([ SG : s ]) , it suffices to consider a fixed subset s ⊆ E ( SG − e ) = E ( SG / e ) . If e joins a component of [ SG − e : s ] to itself, then we define e m to be f SG / e ◦ id ◦ f − SG − e , where f SG / e and f SG − e are similarly defined as f in Definition 4.5. Otherwise, wedefine e m to be f SG / e ◦ m ◦ f − SG − e , where m denotes the multiplication m : M ⊗ M → M . One caneasily check that e m is well defined.We claim that the complex C • ( SG ) is the mapping cone of e m : C • ( SG − e ) → C • ( SG / e ) .Notice that C i ( SG ) = L s ⊆ E ( SG ) , | s | = i M s ( SG ) = L e / ∈ s ⊆ E ( SG ) , | s | = i M s ( SG ) L L e ∈ s ⊆ E ( SG ) , | s | = i M s ( SG ) ,and there is an obvious isomorphism between C i ( SG − e ) = L s ⊆ E ( SG − e ) , | s | = i M s ( SG − e ) and L e / ∈ s ⊆ E ( SG ) , | s | = i M s ( SG ) .For the second summand L e ∈ s ⊆ E ( SG ) , | s | = i M s ( SG ) , one observes that for any s ∈ E ( SG / e ) , there isa one-to-one correspondence between the components of [ SG / e : s ] and the components of [ SG : s ∪ { e } ] . Since the sign of e is positive, then a component of [ SG / e : s ] is balanced if and only if thecorresponding component in [ SG : s ∪ { e } ] is balanced. It follows that L e ∈ s ⊆ E ( SG ) , | s | = i M s ( SG ) ∼ = L s ⊆ E ( SG / e ) , | s | = i − M s ( SG / e ) = C i − ( SG / e ) ,and hence C i ( SG ) = C i ( SG − e ) L C i − ( SG / e ) .Consider the differential d ′ s : C i ( SG ) → C i + ( SG ) given by (cid:18) d e m − d (cid:19) , where d denotes thedifferential on C • ( SG − e ) and d denotes the differential on C • ( SG / e ) . Recall that e is the firstedge, it is not difficult to find that d ′ s coincides with the differential d s defined in Proposition 4.6.This finishes the proof of the claim and the result follows directly. (cid:3) Corollary 4.10.
Let SG be a signed graph and e a positive edge, we have the following long exact sequence · · · → H i − ( SG / e ) → H i ( SG ) → H i ( SG − e ) → H i ( SG / e ) → · · · . CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 11
Remark . It would be helpful to describe the two maps α ∗ : H i − ( SG / e ) → H i ( SG ) and β ∗ : H i ( SG ) → H i ( SG − e ) intuitively. In order to understand α ∗ , it suffices to consider the map α : C i − ( SG / e ) → C i ( SG ) . With a given enhanced state S = ( s , c ) of SG / e , notice that [ SG / e : s ] and [ SG : s ∪ { e } ] not only have the same number of components but also the same number of balancedcomponents, since e is positive. The we can define α ( S ) = ( s ∪ { e } , c e ) , which induces the map α ∗ : H i − ( SG / e ) → H i ( SG ) . For β ∗ , let us choose an enhanced state S = ( s , c ) of SG . If e / ∈ s , thenwe define β ( S ) = S , which is also an enhanced state of SG − e , since removing e from a balancedcomponent yields a balanced component. Otherwise, if e ∈ s then we set β ( S ) =
0. The map β ∗ : H i ( SG ) → H i ( SG − e ) can be induced from β .4.3. A categorification for the balanced chromatic polynomial.
In this subsection, we discuss howto categorify the balanced chromatic polynomial, i.e. the signed chromatic polynomial with even λ .4.3.1. Cochain groups on signed graphs.
Let SG be a signed graph on G , for each signed subgraph [ SG : s ] led by s ⊆ E ( SG ) , we assign a graded Z -module M bs ( SG ) as follows: • if [ SG : s ] is balanced, we assigned a copy of M to each component and then take tensorproduct, i.e. M bs ( SG ) = M ⊗ k ( s ) if [ SG : s ] is balanced. • if [ SG : s ] is unbalanced, we assigned 0 to it.In this case, it is guaranteed that q dim M bs ( SG ) = Q [ SG : s ] ( + q ) when q is odd. As an analogy ofProposition 4.3, we have the following result. Proposition 4.12.
Let SG be a signed graph, we define C ib ( SG ) = L s ⊆ E ( G ) , | s | = i M bs ( SG ) , thenP SG ( + q ) = ∑ i ≥ ( − ) i ∑ s ⊆ E ( G ) , | s | = i qdimM bs ( SG ) = ∑ i ≥ ( − ) i qdimC ib ( SG ) , if q is an odd integer. The differential.
Definition 4.13.
An enhanced state S = ( s , c ) of G is an enhanced state of SG for balanced chromaticpolynomial if and only if [ G : s ] is balanced.As before, we set C i , jb ( SG ) = span h S | S is an enhanced state of SG with i ( S ) = i , j ( S ) = j i ,where i ( S ) = | s | and j ( S ) equals the number of components that x is assigned to. It follows imme-diately that C ib ( SG ) = L j ≥ C i , jb ( SG ) .Now we introduce a map f b : C i , j ( G ) → C i , jb ( SG ) for balanced chromatic polynomial. Definition 4.14.
Let SG be a signed graph on G , for each enhanced state S = ( s , c ) in C i , j ( G ) , wedefine f b ( S ) = ( S , [ SG : s ] is balanced;0, otherwise.We extend f b to a linear projection from C i , j ( G ) to C i , jb ( SG ) , or, from C i ( G ) to C ib ( SG ) if one sumsover j . Let us still use f b to denote it.By using f b , we define the differential d b as the below, which is similar to the definition of d s . C i ( G ) d / / f b (cid:15) (cid:15) C i + ( G ) f b (cid:15) (cid:15) C ib ( SG ) d b / / C i + b ( G ) Proposition 4.15.
The map d b : C ib ( SG ) → C i + b ( SG ) defined by d b = f b ◦ d ◦ f − b is well defined.Proof. Similar to the proof of Proposition 4.6, it suffices to show that for any z = m ∑ i = a i S i ( a i = ) ,if f b ( z ) = f b ◦ d ( z ) =
0. Here S i = ( s i , c i ) for some s i ⊂ E ( SG ) . According to the definitionof f b , [ SG : s i ] must be unbalanced. Notice that the differential d corresponds to the following twocases(1) adding an edge connecting an unbalanced component to itself,(2) adding an edge connecting an unbalanced component with another component.Both of them give rise to a new unbalanced component. It follows that f b ◦ d ( S i ) = f b ◦ d ( z ) = (cid:3) Since d = d b = f b ◦ d ◦ f − b , we obtain d b = f b ◦ d ◦ f − b = C b ( SG ) d b −→ C b ( SG ) d b −→ C b ( SG ) d b −→ · · · d b −→ C ib ( SG ) d b −→ · · · . Definition 4.16.
We call the cohomology groups of the cochain complex above the balanced chromaticcohomology groups of SG and denote them by H • b ( SG ) .Similar to Proposition 4.8, the balanced chromatic cohomology groups H • b ( SG ) are independentof the choice of the order of edges. Now we give two examples. The first one, as a supplement ofExample 4.2, calculates the balanced chromatic cohomology groups of the signed graph SP . Thesecond one is devoted to calculate the balanced chromatic cohomology groups of SP , see Figure 4. v v e , + e , − F IGURE
4. An unbalanced signed graph SP Example . We use the same notions as that in Example 4.2. It’s easy to find that for each i ∈{
0, 1, 2 } , we have C ib ( SP ) = C i ( SP ) , B ib ( SP ) = B i ( SP ) and C b ( SP ) = B b ( SP ) =
0. On theother hand, Z b ( SP ) = Z ( SP ) , Z b ( SP ) = Z ( SP ) , Z b ( SP ) = C ( SP ) , Z b ( SP ) =
0. Then weobtain the balanced chromatic cohomology groups as below H b ( SP ) = Z ( SP ) / B ( SP ) = H ( SP ) ∼ = Z { } ; H b ( SP ) = Z ( SP ) / B ( SP ) = H ( SP ) ∼ = Z { } ⊕ Z { } ; H b ( SP ) = Z ( SP ) / B ( SP ) ∼ = Z { } ⊕ Z ; H b ( SP ) = SGP SP ( + q ) = ∑ i ≥ ( − ) i q dim C ib ( SP ) = ∑ i ≥ ( − ) i q dim H ib ( SP ) = q − q + q + = q + q is odd. This result coincides with the result obtained in Example 3.2. Example . Let SP be an unbalanced signed graph on P , where E ( SP ) = { e , e } and σ ( e ) = + σ ( e ) = −
1, as Figure 4 shows. For each enhanced state S = ( s , c ) of SP , we denote s and c similarly CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 13 to Example 4.2, then we can calculate the cochain groups and the cohomology groups as follows. C b ( SP ) = span h ( xx ) , (
00, 1 x ) , ( x ) , (
00, 11 ) i ; C b ( SP ) = span h ( xx ) , ( xx ) , (
01, 11 ) , (
10, 11 ) i ; C b ( SP ) = B b ( SP ) = B b ( SP ) = span h ( xx ) + ( xx ) , (
10, 11 ) + (
01, 11 ) i ; B b ( SP ) = Z b ( SP ) = span h ( xx ) , (
00, 1 x ) − ( x ) i ; Z b ( SP ) = span h ( xx ) , ( xx ) , (
01, 11 ) , (
10, 11 ) i ; Z b ( SP ) = H b ( SP ) = Z b ( SP ) / B b ( SP ) ∼ = Z { } ⊕ Z { } ; H b ( SP ) = Z b ( SP ) / B b ( SP ) ∼ = Z { } ⊕ Z ; H b ( SP ) = Z b ( SP ) / B b ( SP ) ∼ = P SG ( + q ) = ∑ i ≥ ( − ) i q dim C ib ( SG ) = ∑ i ≥ ( − ) i q dim H ib ( SG ) = q + q − q − = q − q is an odd integer. And we can check this result intuitively: suppose there are 1 + q colors,if we assign one color to v , to which there are 1 + q approaches. There are q − v . According to the multiplication principle, P SP ( + q ) = ( q + )( q − ) = q − q is odd.We remark that it was proved that for unsigned graphs, the knight move conjecture holds forthe chromatic cohomology with rational coefficients [3]. More precisely, they proved that for aunsigned graph G , the nontrivial cohomology groups come in isomorphic pairs: H i , | V ( G ) |− i ( G ; Q ) ∼ = H i + | V ( G ) |− i − ( G ; Q ) . According to the examples above, we find that this is not true for signedchromatic cohomology.4.3.3. Another long exact sequence.
As an analogy to Corollary 4.10, the deletion-contraction relationfor balanced chromatic polynomial also corresponds to a long exact sequence. This can be derivedfrom the following theorem.
Theorem 4.17.
Let SG be a signed graph and e a positive edge, we have the following short exact sequence → C •− b ( SG / e ) → C • b ( SG ) → C • b ( SG − e ) → .Proof. The proof is a mimic of that of Theorem 4.9. For cochain groups, we still have C ib ( SG ) = L e / ∈ s ⊆ E ( SG ) , | s | = i M bs ( SG ) ⊕ L e ∈ s ⊆ E ( SG ) , | s | = i M bs ( SG ) = C ib ( SG − e ) ⊕ C i − b ( SG / e ) ,where e is set as the first edge. The morphism f m b : C • b ( SG − e ) → C • b ( SG / e ) can be similarly definedas f m b = ( f b ◦ id ◦ f − b , if e connects one component to itself f b ◦ m ◦ f − b , if e connects two different componentswhere m denotes the multiplication on M and f b is the map introduced in Definition 4.14. Let us use d b and d b to denote the differential maps on C • ( SG − e ) and C • ( SG / e ) respectively. It is not difficultto check that the differential d ′ b = (cid:18) d b f m b − d b (cid:19) defined on C • b ( SG − e ) L C •− b ( SG / e ) coincideswith the differential d b defined on C • b ( SG ) . In other words, the complex C • b ( SG ) is the mappingcone of f m b : C • b ( SG − e ) → C • b ( SG / e ) . The result follows. (cid:3) Corollary 4.18.
Let SG be a signed graph and e a positive edge, we have the following long exact sequence · · · → H i − b ( SG / e ) → H ib ( SG ) → H ib ( SG − e ) → H ib ( SG / e ) → · · · .
5. S
OME PROPERTIES
Relation to the chromatic cohomology groups of unsigned graphs.
In Section 4, we intro-duced two cochain complexes for signed graphs based on the categorification of the chromatic poly-nomial of unsigned graphs, which was proposed by Laure Helme-Guizon and Yongwu Rong in [5].So it’s natural to consider the relation among the cohomology groups of these three cochain com-plexes. Obviously, if all the edges of a signed graph are positive, then all these cohomology groupscoincide. This conclusion can be enhanced a little bit as follows.For a signed graph SG , let us use n b ( SG ) to denote the length of the shortest unbalanced circuits.In other words, any [ SG : s ] is balanced provided that | s | ≤ n b ( SG ) − Proposition 5.1.
For any i ≤ n b − , we have H i ( SG ) = H i ( G ) = H ib ( SG ) .Proof. For any i ≥ n b , by replacing all the cochain groups C i ( G ) , C i ( SG ) and C ib ( SG ) with 0, oneobtains three new cochain complexes. Since there is no negative circuit now, these three cochaincomplexes are exactly the same, hence the cohomology groups are isomorphic mutually. (cid:3) Disjoint union of two signed graphs.
Let SG and SG be two signed graphs, we denote theirdisjoint union by SG ⊔ SG . On the chain complex level, we have C • ( SG ⊔ SG ) = C • ( SG ) ⊗ C • ( SG ) and C • b ( SG ⊔ SG ) = C • b ( SG ) ⊗ C • b ( SG ) . As a corollary of the K ¨unneth theorem, thecohomology groups of SG , SG and their disjoint union satisfy the following relations. Proposition 5.2.
For each i ∈ N , we haveH i ( SG ⊔ SG ) ∼ = [ ⊕ p + q = i H p ( SG ) ⊗ H q ( SG )] ⊕ [ ⊕ p + q = i + H p ( SG ) ∗ H q ( SG )] , H ib ( SG ⊔ SG ) ∼ = [ ⊕ p + q = i H pb ( SG ) ⊗ H qb ( SG )] ⊕ [ ⊕ p + q = i + H pb ( SG ) ∗ H qb ( SG )] , where ∗ is the torsion product of abelian groups. In particular, when SG is a trivial graph, i.e. the graph with exactly one vertex, it is easy to findthat H i ( SG ) = H ib ( SG ) = Z ⊕ Z x . It follows that H i ( SG ⊔ SG ) ∼ = H i ( SG ) ⊗ ( Z ⊕ Z x ) , H ib ( SG ⊔ SG ) ∼ = H ib ( SG ) ⊗ ( Z ⊕ Z x ) .5.3. Vertex switching operation.
Recall that two signed graphs are equivalent if they are related byseveral vertex switchings. As we mentioned before, equivalent signed graphs have the same signedchromatic polynomial. The following result tells us that vertex switching not only preserves thesigned chromatic polynomial, but also the signed chromatic cohomology groups.
Proposition 5.3.
If two signed graphs SG , SG are equivalent, then H i ( SG ) ∼ = H i ( SG ) and H ib ( SG ) ∼ = H ib ( SG ) .Proof. It suffices to consider the case that SG is obtained from SG by applying vertex switchingon a vertex v ∈ V ( SG ) . Notice that for any s ⊆ E ( SG ) , a component in [ SG : s ] is balanced ifand only if the corresponding component in [ SG : s ] is also balanced. This induces a cochain mapfrom C • ( SG ) to C • ( SG ) and another cochain map from C • b ( SG ) to C • b ( SG ) , both of which induceisomorphisms between the cohomology groups. (cid:3) Contracting a pendant edge.Definition 5.4.
Let SG be a signed graph, suppose v ∈ V ( SG ) is a vertex of degree one. We call theedge incident with v a pendant edge of SG .For a given signed graph SG and a positive pendant edge e (one can apply vertex switching on v if necessary), Proposition 3.1 tells us that CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 15 P SG ( λ ) = P SG − e ( λ ) − P SG / e ( λ ) = ( λ − ) P SG / e ( λ ) .The following proposition can be seen as a categorification of this. Proposition 5.5.
Let e be a pendant edge in a signed graph SG. For each i, we have H i ( SG ) ∼ = H i ( SG / e ) { } and H ib ( SG ) ∼ = H ib ( SG / e ) { } .Proof. We only prove H i ( SG ) ∼ = H i ( SG / e ) { } , the balanced version can be proved analogously.By switching v if necessary, we assume the sign of e is positive and e is the first edge. The keyobservation is, for any s ⊂ E ( SG ) the pendant edge e has no effect on the balanced components of [ SG : s ] . The main idea of the proof is similar to the unsigned case [5]. We sketch the outline here.According to Corollary 4.10, we have the following long exact sequence · · · → H i − ( SG / e ) → H i ( SG ) → H i ( SG − e ) → H i ( SG / e ) → · · · .On the other hand, since SG − e = SG / e ∪ { v } , Proposition 5.2 tells us that H i ( SG − e ) ∼ = H i ( SG / e ) ⊗ ( Z ⊕ Z x ) ∼ = H i ( SG / e ) ⊕ H i ( SG / e ) { } .By identifying H i ( SG − e ) with H i ( SG / e ) ⊕ H i ( SG / e ) { } , it suffices to show that the map γ ∗ : H i ( SG / e ) ⊕ H i ( SG / e ) { } → H i ( SG / e ) sends ( x , 0 ) to x .In fact, for any x = [ ∑ i a i S i ] ∈ H i ( SG / e ) , where S i = ( s i , c i ) , we extend each S i to be an enhancedstate in C i ( SG − e ) by adding an isolated vertex v with color 1. Then the map γ ∗ sends each [( s i , c i )] to [( s i ∪ { e } ) / e , ( c i ) e ] . Notice that adding a positive pendant edge preserves the balance of eachcomponent. On the other hand, since v is colored by 1 and multiplication by 1 is just the identitymap, it follows that γ ∗ (( x , 0 )) = x .Therefore γ ∗ is surjective and hence the long exact sequence splits into infinitely many short exactsequences 0 → H i ( SG ) → H i ( SG / e ) ⊕ H i ( SG / e ) { } γ ∗ → H i ( SG / e ) → H i ( SG ) ∼ = H i ( SG / e ) { } . (cid:3) Loops and parallel edges.
We first discuss the effect of positive/negative loops on the signedchromatic cohomology. Similar to the fact that signed graphs with positive loops have zero signedchromatic polynomial, positive loops also kill the signed chromatic cohomology.
Proposition 5.6.
If a signed graph SG has a positive loop e, then H i ( SG ) = H ib ( SG ) = .Proof. The assumption e is a positive loop implies that SG / e = SG − e . By investigating the map γ ∗ in the following long exact sequence · · · → H i − ( SG / e ) → H i ( SG ) → H i ( SG − e ) γ ∗ → H i ( SG / e ) → H i + ( SG ) → · · · ,it is not difficult to find that γ ∗ is an isomorphism. We conclude that H i ( SG ) =
0. The unbalancedcase can be proved similarly. (cid:3)
Proposition 5.7.
If a signed graph SG has a negative loop e, then H ib ( SG ) = H ib ( SG − e ) .Proof. Since e is negative, Corollary 4.18 does not work here. However, if we go back to the cochaincomplex, we have the following decomposition C ib ( SG ) = L e / ∈ s ⊆ E ( SG ) , | s | = i M bs ( SG ) ⊕ L e ∈ s ⊆ E ( SG ) , | s | = i M bs ( SG ) .If a subset s ⊆ E ( SG ) includes the negative loop e , then [ SG : s ] is unbalanced, therefore the as-sociated M bs ( SG ) =
0. It follows that the second summand above vanishes and the result followsimmediately. (cid:3)
Now we turn to discuss the effect of parallel edges on signed chromatic cohomology. Recall thattwo edges join the same pair of vertices, then these two edges are called parallel edges . Here we allowthe two endpoints coincide with each other.
Proposition 5.8.
Let SG be a signed graph, and e , e ′ ∈ E ( SG ) are a pair of parallel edges with the same sign.Then we have H i ( SG ) = H i ( SG − e ′ ) and H ib ( SG ) = H ib ( SG − e ′ ) . In other words, the signed chromaticcohomology groups are unchanged if one replaces the parallel edges with the same sign by a single one.Proof. If both e and e ′ are positive, we divide our discussion into two cases. • The two endpoints of e and e ′ coincide, i.e. both e and e ′ are positive loops. In this case, theresult follows from Proposition 5.6. • The two endpoints of e and e ′ are distinct. Then e becomes a positive loop in SG / e ′ , hencewe obtain H i ( SG / e ′ ) = H ib ( SG / e ′ ) =
0. It follows from the two long exact sequences that H i ( SG ) = H i ( SG − e ′ ) and H ib ( SG ) = H ib ( SG − e ′ ) .If both e and e ′ are negative, there are also two situations. • Both e and e ′ are not loops. By switching one of the two endpoints of e and e ′ we obtain twopositive parallel edges, which has been discussed above. • Both e and e ′ are loops. The balanced case H ib ( SG ) = H ib ( SG − e ′ ) follows directly fromProposition 5.7. The rest of the proof is devoted to show that H i ( SG ) = H i ( SG − e ′ ) . Sup-pose both e and e ′ connects v ∈ V ( SG ) to itself. We define a new signed graph SG ′ bysplitting v into two vertices, say v , v , and adding a new positive edge e which connects v and v . All edges incident to v in SG are now incident to v in SG ′ . See Figure 5. SG = v e , − e ′ , − , SG ′ = v v e , + e , − e ′ , − F IGURE
5. Adding a new vertexNow we have SG = SG ′ / e , SG − e ′ = ( SG ′ − e ′ ) / e and e , e ′ are negative edges con-necting v and v in SG ′ and SG ′ − e . According to our previous discussion, if we switch v , delete e ′ and then switch v again, these operations induce isomorphisms H i ( SG ′ ) ∼ = H i ( SG ′ − e ′ ) and H i ( SG ′ − e ) ∼ = H i ( SG ′ − e − e ′ ) . The homomorphism δ ∗ : H i ( SG ′ / e ) → H i (( SG ′ − e ′ ) / e ) in the commutative diagram below can be defined as follows. Given anenhanced state S = ( s , c ) of SG ′ / e , if e ′ / ∈ s , then we define δ ( S ) = S , which is also an en-hanced state of ( SG ′ − e ′ ) / e . Otherwise, we define δ ( S ) =
0. This map makes the diagrambelow commutative and induces a homomorphism δ ∗ : H i ( SG ′ / e ) → H i (( SG ′ − e ′ ) / e ) . H i ( SG ′ ) ∼ = (cid:15) (cid:15) / / H i ( SG ′ − e ) ∼ = (cid:15) (cid:15) / / H i ( SG ′ / e ) δ ∗ (cid:15) (cid:15) / / H i + ( SG ′ ) ∼ = (cid:15) (cid:15) / / H i + ( SG ′ − e ) ∼ = (cid:15) (cid:15) H i ( SG ′ − e ′ ) / / H i ( SG ′ − e ′ − e ) / / H i (( SG ′ − e ′ ) / e ) / / H i + ( SG ′ − e ′ ) / / H i + ( SG ′ − e ′ − e ) We conclude that H i ( SG ) ∼ = H i ( SG ′ / e ) ∼ = H i (( SG ′ − e ′ ) / e ) ∼ = H i ( SG − e ′ ) ,the fact that δ ∗ is an isomorphism is derived from the five lemma. The proof is finished. (cid:3) CATEGORIFICATION FOR THE SIGNED CHROMATIC POLYNOMIAL 17
Trees and polygon graphs.
As an application of the properties discussed above, we describethe signed chromatic cohomology groups for some classes of signed graphs.
Example . Let SN nm be the signed graph with m vertices, on n of which there is a negative loop.When m = n =
1, we can calculate the signed chromatic cohomology groups for SN as follows. C ( SN ) = span h (
0, 1 ) , ( x ) i C ( SN ) = span h (
1, 1 ) i B ( SN ) = B ( SN ) = span h (
1, 1 ) i Z ( SN ) = span h ( x ) i Z ( SN ) = span h (
1, 1 ) i H ( SN ) = Z ( SN ) / B ( SN ) ∼ = Z x H ( SN ) = Z ( SN ) / B ( SN ) ∼ = C b ( SN ) = span h (
0, 1 ) , ( x ) i C b ( SN ) = B b ( SN ) = B b ( SN ) = Z b ( SN ) = span h (
0, 1 ) , ( x ) i Z b ( SN ) = H b ( SN ) = Z b ( SN ) / B b ( SN ) ∼ = Z ⊕ Z x H b ( SN ) = Z b ( SN ) / B b ( SN ) ∼ = H i ( SN nm ) ∼ = ( ( Z ⊕ Z { } ) ⊗ ( m − n ) ⊗ ( Z { } ) ⊗ n , i = i ≥ H ib ( SN nm ) ∼ = ( ( Z ⊕ Z { } ) ⊗ m , i = i ≥ H ib ( SN nm ) ∼ = H ib ( SN m ) also can be deduced from Proposition 5.7. Example . Let ST n = ( T n , σ ) , where T n is a tree with n edges. By repeatedly using Proposition 5.5one obtains H i ( ST n ) ∼ = H ib ( ST n ) ∼ = ( Z { n } ⊕ Z { n + } , i = i ≥ Example . Let SP n = ( P n , σ ) be an unbalanced polygon graph with n edges. When n =
1, the case SP = SN has been discussed in Example 5.1. For this reason, next let us assume n ≥ P n arenot unique, they are all equivalent. In order to see this, first notice that two unbalanced polygongraphs with only one negative edge are equivalent. If an unbalanced polygon graph has more thanthree negative edges, choose two of them such that we can find a positive path connecting them.By applying vertex switching on the endpoints of this positive path we obtain a new unbalancedpolygon graph with two negative edges less.We label the vertices of SP n by v , v , · · · , v n monotonically so that each v i is adjacent to v i ± ( ≤ i ≤ n ) , where v = v n and v n + = v . Let e be a positive edge connecting v and v n , the SP n / e = SP n − and SP n − e is a tree. Then we have the following long exact sequence · · · → H i − ( SP n − e ) → H i − ( SP n / e ) → H i ( SP n ) → H i ( SP n − e ) → · · · .As SP n − e is a tree, we have H i ( SP n − e ) = i ≥
1. Thus for any i ≥
2, we have H i ( SP n ) ∼ = H i − ( SP n / e ) ∼ = H i − ( SP n − ) , and it follows that H i ( SP n ) ∼ = ( H ( SP n − i + ) , if 2 ≤ i ≤ n ;0, if i > n .Then for each n ≥ H n ( SP n ) ∼ = H ( SP ) ∼ = H n − ( SP n ) ∼ = H ( SP ) ∼ = Z { } , which has beencalculated in Example 4.2. On the other hand, it has been calculated in [5] thatFor i = H ( P n ) ∼ = Z { n } ⊕ Z { n − } if n is even and n ≥ Z { n } if n is odd and n ≥ n = i > H i ( P n ) ∼ = Z { n − i } ⊕ Z { n − i − } if n − i ≥ n is even; Z { n − i } if n − i ≥ n is odd;0 if n − i ≤ H i ( SP n ) = H i ( P n ) for all i ≤ n −
2, then we obtain all thecohomology groups.For i = H ( SP n ) ∼ = Z { n } ⊕ Z { n − } if n is even and n ≥ Z { n } if n is odd and n ≥ Z { } if n = i > H i ( SP n ) ∼ = Z { n − i } ⊕ Z { n − i − } if i ≤ n − n is even; Z { n − i } if i ≤ n − n is odd; Z { } if i = n − i ≥ n .We work out the balanced cohomology groups parallelly.For i = H b ( SP n ) ∼ = Z { n } ⊕ Z { n − } if n is even and n ≥ Z { n } if n is odd and n ≥ Z { } ⊕ Z if n = i > H ib ( P n ) ∼ = Z { n − i } ⊕ Z { n − i − } if i ≤ n − n is even; Z { n − i } if i ≤ n − n is odd; Z { } ⊕ Z if i = n − i ≥ n .A CKNOWLEDGEMENTS
ZY Cheng is supported by NSFC 11771042 and NSFC 12071034. ZY Lei, YT Wang and YG Zhangare supported by an undergraduate research project of Beijing Normal University.R
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Concepts of signed graph coloring , European Journal of Combinatorics (2021).[17] T. Zalavsky, Signed graph coloring , Discrete Mathematics (1982), 215–228.L ABORATORY OF M ATHEMATICS AND C OMPLEX S YSTEMS , S
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