A bivariate chromatic polynomial for signed graphs

A bivariate chromatic polynomial for signed graphs

Abstract.

We study Dohmen–Pönitz–Tittmann’s bivariate chromatic polynomial which counts all -colorings of a graph such that adjacent vertices get different colors if they are . Our first contribution is an extension of to signed graphs, for which we obtain an inclusion–exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for and its signed-graph analogues, reminiscent of Stanley’s reciprocity theorem linking chromatic polynomials to acyclic orientations.

Key words and phrases:
Signed graph, bivariate chromatic polynomial, deletion–contraction, combinatorial reciprocity, acyclic orientation, graphic arrangement, inside-out polytope.
2000 Mathematics Subject Classification:
Primary 05C15; Secondary 05A15, 05C22, 11P21, 52C35.
We thank the referees for helpful suggestions. This research was partially supported by the U. S. National Science Foundation through the grants DMS-1162638 (Beck) and DGE-0841164 (Hardin).

1. Introduction

Graph coloring problems are ubiquitous in many areas within and outside of mathematics. For a positive integer , let and . We study the bivariate chromatic polynomial of a graph , first introduced in [5] and defined as the counting function of colorings that satisfy for any edge

The usual chromatic polynomial of can be recovered as the special evaluation . Dohmen, Pönitz, and Tittmann provided basic properties of in [5], including polynomiality and special evaluations yielding the matching and independence polynomials of . Subsequent results include a deletion–contraction formula and applications to Fibonacci-sequence identities [9], common generalizations of and the Tutte polynomial [1], and closed formulas for paths and cycles [4].

Our first goal is to introduce and study the natural analogue of the bivariate chromatic polynomial for signed graphs, which originated in the social sciences and have found applications also in biology, physics, computer science, and economics; see [14] for a comprehensive bibliography. A signed graph consists of a graph and a signature . The underlying graph may have multiple edges and, besides the usual links and loops, also halfedges (with only one endpoint) and loose edges (no endpoints); the latter are irrelevant for coloring questions, and so we assume in this paper that has no loose edges. An unsigned graph can be realized by a signed graph all of whose edges are labelled with .

We define the function as counting the proper -colorings , namely, those colorings that satisfy for any edge

This bivariate chromatic polynomial (in Corollary 5 we will see that is indeed a polynomial) specializes to Zaslavsky’s chromatic polynomial of signed graphs [11] in the case . As in Zaslavsky’s theory, comes with a companion, the zero-free bivariate chromatic polynomial which counts all proper -colorings .

Our first result is a deletion–contraction formula, the common generalization of [11, Theorem 2.3] and [9, Lemma 1.1]. The definitions of deletions and contractions of signed graphs are reviewed in detail in Section 2, where we also prove our other results for the bivariate chromatic polynomials.

Theorem 1.

Let be a signed graph. If is a halfedge or negative loop then

if is not a halfedge or negative loop then

and

where is the vertex to which contracts in .

A component of the signed graph is balanced if it contains no halfedges and each cycle has positive sign product, and it is antibalanced if its negative is balanced. We define the antibalance polynomial of as

where denotes the number of components of . This polynomial relates to the zero-free bivariate chromatic polynomial as follows:

Theorem 2.

Our second goal is to prove reciprocity theorems for the bichromatic polynomials for graphs and signed graphs, in analogy with the following theorem of Stanley [10] on the usual chromatic polynomial . We call an orientation of a graph compatible with the coloring if for any edge oriented from to .

Theorem 3 (Stanley).

For , equals the number of -colorings of , each counted with multiplicity equal to the number of compatible acyclic orientations of . In particular, equals the number of acyclic orientations of .

Our generalization for bivariate chromatic polynomials is as follows.

Theorem 4.

For and , equals the number of -colorings of , each counted with multiplicity: a -coloring has multiplicity equal to the number of compatible acyclic orientations of , and a coloring that uses at least one color has multiplicity .

We prove this theorem in Section 3, where we also give an analogous reciprocity theorem for the bivariate chromatic polynomials of signed graphs. We finish with a few open problems in Section 4.

2. Bivariate Chromatic Polynomials for Signed Graphs

We first review a few constructs on a signed graph . The restriction of to an edge set is the signed graph . For , we denote by (the deletion of ) the restriction of to . For , denote by the restriction of to where is the set of all edges incident to .

Switching by results in the new signed graph where . Switching does not alter balance, and any balanced signed graph can be obtained from switching an all-positive graph [12]. We also note that there is a natural bijection of colorings of and a switched version of it, and this bijection preserves the number of proper -colorings.

The contraction of by , denoted by , is defined as follows [12]: switch so that every balanced component of is all positive, coalesce all vertices of each balanced component, and discard the remaining vertices and all edges in ; note that this may produce halfedges. For example, if for a link , is obtained by switching so that and then contracting as in the case of unsigned graphs, that is, disregard and identify its two endpoints. As a second example, if is a negative loop at , then has vertex set and edge set resulting from by deleting and converting all edges incident with to half edges.

Before proving Theorem 1, we give two illustrating examples. First, a signed path with three vertices (Figure 1):

Figure 1. Deletion–contraction at a halfedge of a signed path .

Our second example is a signed 3-cycle (Figure 2):

Figure 2. Deletion–contraction at an edge of a signed 3-cycle .
Proof of Theorem 1.

Let be a halfedge or negative loop of , and let be its incident vertex. Then cannot be colored zero, and so we have to subtract from the colorings of those which color zero (which are in bijection with the colorings of ).

Now let be an edge of that is not a halfedge or negative loop. We have to subtract from the colorings of those which color the endpoints of the same (which are in bijection with the colorings of ) and add back in those where the latter color is (the number of which is times the number of proper -colorings of ). ∎

By induction on the number of edges of a signed graph, we immediately conclude:

Corollary 5.

The chromatic counting functions and are polynomials in and .

Our next result is the signed-graph analog of [5, Theorem 1].

Theorem 6.

Let be a signed graph. Then

Naturally, the polynomials in the summations should be interpreted as Zaslavsky’s chromatic polynomials.

Proof.

Every proper -coloring of can be obtained by first choosing a subset of that is colored with colors ; there are such colorings for these vertices. The remaining subgraph has to be colored properly using colors . ∎

The above proof is virtually identical to that of [5, Theorem 1], and thus we obtain, as an analogous consequence, the following corollary, paralleling [5, Corollary 2], regarding the independence polynomial

(Here is independent if no two vertices in are adjacent.)

Corollary 7.

Proof.

By Theorem 6, we have

Now note that equals one if is independent and zero otherwise. ∎

Proof of Theorem 2.

By Theorem 6,

Now equals if is antibalanced, and if is not antibalanced. Thus

For completeness sake, we state the signed analogue of [5, Theorem 3]; its proof is virtually identical to the unsigned case.

Theorem 8.

 and  

3. Bivariate Chromatic Reciprocity Theorems

The proofs of our reciprocity theorems follow along the lines of the proof of Stanley’s Theorem 3 given in [3], which introduced the general setup of an inside-out polytope consisting of a rational polytope and a rational hyperplane arrangement in ; that is, the linear equations and inequalities defining and have integer coefficients. (The proper understanding of this section assumes familiarity with [3].) The goal is to compute the counting function

and it follows from Ehrhart’s theory of counting lattice points in dilates of rational polytopes [2, 6] that this function is a quasipolynomial in whose degree is , whose (constant) leading coefficient is the normalized lattice volume of , and whose period divides the lcm of all denominators that appear in the coordinates of the vertices of . In our case, all vertices of will be integral, so that the resulting counting functions will be polynomials. Furthermore, [3] established the reciprocity theorem

(1)

where and denote the interior and closure of , respectively, and

(2)

where denotes the number of closed regions of containing . (A region of is a connected component of ; a closed region is the closure of a region.) See [3] for this and several more properties of inside-out polytopes. The concept of inside-out polytopes has been applied to a number of combinatorial settings; at the heart of any such application is an interpretation of the regions of ; from this point of view, Stanley’s Theorem 3 follows from Greene’s observation [7, 8] that the regions of the graphic arrangement (for a given graph )

in are in one-to-one correspondence with the acyclic orientations of .

Proof of Theorem 4.

Given , let be the unit cube in and

i.e., is the difference of two evaluations (at and ) of the Ehrhart polynomial of . By Ehrhart–Macdonald reciprocity (see, e.g., [2, Chapter 4]),

(3)

On the other hand, it is natural to interpret the bichromatic polynomial of geometrically as

(see Figure 3 for an illustrative example).

Figure 3. The proper -colorings of with and .

Thus, by (1) and (3),

What we are counting on the right-hand side are the integer lattice points in the cube , with multiplicity equal to if outside the cube , otherwise with multiplicity equal to the number of closed regions of the points lies in. As we mentioned above (and as was used in [3]), the latter can be interpreted as the number of compatible acyclic orientations of . It is now a short step to re-interpret the lattice points in as -colorings and the ones in as -colorings of . ∎

In order to state and prove the analogous reciprocity theorem for bichromatic polynomials for signed graphs, we need more definitions. An orientation of a signed graph is obtained from a bidirection of the underlying graph , where the endpoints of each edge are independently oriented, in such a way that the two arrows on an edge point in the same direction if and they conflict if . We express the bidirection (and hence the orientation) by means of an incidence function defined on the edge ends: if the arrow on at points into , and if it points away from ; with this definition we obtain for an edge . (See [13] for more details.)

Following [11], we call a coloring and an orientation compatible if for any link

and for any halfedge or negative loop at

Furthermore, an orientation is acyclic if no cycle has a source or sink (i.e., a vertex for which both incident edges point either into or away from ). Zaslavsky [11] proved the following analogue of Stanley’s Theorem 3 for the chromatic polynomial of a signed graph :

Theorem 9 (Zaslavsky).

For , equals the number of -colorings of , each counted with multiplicity equal to the number of compatible acyclic orientations of . In particular, equals the number of acyclic orientations of .

Our analogue of this theorem for bivariate chromatic polynomials is as follows.

Theorem 10.

For and , equals the number of -colorings of , each counted with multiplicity: a -coloring has multiplicity equal to the number of compatible acyclic orientations of , and a coloring that uses at least one color with absolute value has multiplicity .

Proof.

We follow the proof of Theorem 4, with a few modifications: Given , let and

By Ehrhart–Macdonald reciprocity,

(4)

To construct an inside-out counting function for , we use the hyperplane arrangement

and so

(see Figure 4).

Figure 4. The lattice points corresponding to -colorings of with and .

Thus, by (1) and (4),

What we are counting on the right-hand side are the integer lattice points in the cube , with multiplicity equal to if outside the cube , otherwise with multiplicity equal to the number of closed regions of the points lies in. As shown in [11] (and again used in [3]), the latter can be interpreted as the number of compatible acyclic orientations of . ∎

4. Open Questions

We finish with two venues for future research. First, one can associate several matroids to a signed graph, most prominently Zaslavsky’s frame matriod and (extended) lift matroid [12, 14]. It is a natural question to ask about possible connections between the Tutte polynomials of these matroids and the bivariate chromatic polynomials. Second, gain and biased graphs are natural generalizations of signed graphs [14], and so another natural question concerns possible extensions of our work to these more general constructs.

References

  1. Ilia Averbouch, Benny Godlin, and J. A. Makowsky, An extension of the bivariate polynomial, European J. Combin. 31 (2010), no. 1, 1–17.
  2. Matthias Beck and Sinai Robins, Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, New York, 2007, Electronically available at http://math.sfsu.edu/beck/ccd.html.
  3. Matthias Beck and Thomas Zaslavsky, Inside-out polytopes, Adv. Math. 205 (2006), no. 1, 134–162, arXiv:math.CO/0309330.
  4. Klaus Dohmen, Closed-form expansions for the bivariate chromatic polynomial of paths and cycles, preprint (arXiv:1201.3886), 2012.
  5. Klaus Dohmen, André Pönitz, and Peter Tittmann, A new two-variable generalization of the chromatic polynomial, Discrete Math. Theor. Comput. Sci. 6 (2003), no. 1, 69–89 (electronic).
  6. Eugène Ehrhart, Sur les polyèdres rationnels homothétiques à  dimensions, C. R. Acad. Sci. Paris 254 (1962), 616–618.
  7. Curtis Greene, Acyclic orientations, Higher Combinatorics (M. Aigner, ed.), NATO Adv. Study Inst. Ser., Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht, 1977, pp. 65–68.
  8. Curtis Greene and Thomas Zaslavsky, On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs, Trans. Amer. Math. Soc. 280 (1983), no. 1, 97–126.
  9. Christopher J. Hillar and Troels Windfeldt, Fibonacci identities and graph colorings, Fibonacci Quart. 46/47 (2008/09), no. 3, 220–224, arXiv:0805.0992.
  10. Richard P. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171–178.
  11. Thomas Zaslavsky, Signed graph coloring, Discrete Math. 39 (1982), no. 2, 215–228.
  12. by same author, Signed graphs, Discrete Appl. Math. 4 (1982), 47–74. Erratum, Discrete Appl. Math. 5 (1983), 248.
  13. by same author, Orientation of signed graphs, European J. Combin. 12 (1991), no. 4, 361–375.
  14. by same author, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin. 5 (1998), Dynamic Surveys 8, 124 pp. (electronic), Electronically available at http://www.math.binghamton.edu/zaslav/Bsg/index.html.
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