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On the location of chromatic zeros of series-parallel graphs

04/21/2022
by   Ferenc Bencs, et al.
University of Amsterdam
0

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, showing density outside the disk {z∈ℂ| |z-1| ≤ 1}, we show density of these zeros in the half plane (q)>3/2 and we show there exists an open region U containing the interval (0,32/27) such that U∖{1} does not contain zeros of the chromatic polynomial of series-parallel graphs. We also disprove a conjecture of Sokal by showing that for each large enough integer Δ there exists a series-parallel graph for which all vertices but one have degree at most Δ and whose chromatic polynomial has a zero with real part exceeding Δ.

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1 Introduction

Recall that the chromatic polynomial of a graph is defined as

where denotes the number of components of the graph . We call a number a chromatic root if there exists a graph such that

About twenty years ago Sokal [11] proved that the set of chromatic roots of all graphs is dense in the entire complex plane. In fact, he only used a very small family of graphs to obtain density. In particular, he showed that the chromatic roots of all generalized theta graphs (parallel compositions of equal length paths) are dense outside the disk . (We denote for and by the closed disk centered at of radius .) Extending this family of graphs by taking the disjoint union of each generalized theta graph with an edge and connecting the endpoints of this edge to all other vertices, he then obtained density in the entire complex plane.

As far as we know it is still open whether the chromatic roots of all planar graphs or even series-parallel graphs are dense in the complex plane. Motivated by this question and Sokal’s result we investigate in the present paper what happens inside the disk for the family of series-parallel graphs. See Section 2 for a formal definition of series-parallel graphs. Our first result implies that the chromatic roots of series-parallel are not dense in the complex plane.

Theorem 1.

There exists an open set containing the open interval such that for any and for all series-parallel graphs .

We note that the interval is tight, as shown in [7, 13]. In fact, Jackson [7] even showed that there are no chromatic zeros in the interval . Unfortunately, we were not able to say anything about larger families of graphs and we leave open as a question whether Theorem 1 is true for the family of all planar graphs for example.

In terms of chromatic zeros of series-parallel graphs inside the disk we have found an explicit condition, Theorem 5 below, that allows us to locate many zeros inside this disk. Concretely, we have the following results.

Theorem 2.

Let . Then there exists arbitrarily close to and a series-parallel graph such that .

This result may be seen as a a variation on Thomassen’s result [13] saying that real chromatic zeros (of not necessarily series-parallel graphs) are dense in .

Another result giving many zeros inside is the following.

Theorem 3.

The set of chromatic zeros of all series-parallel graphs is dense in the set

After inspecting our proof of Theorem 3 (given in Section 4) it is clear that one can obtain several strengthenings of this result. Figure 1 below shows a computer generated picture displaying where chromatic zeros of series-parallel graphs can be found as well as the zero-free region from Theorem 1.

Figure 1: A pixel-picture of chromatic zeroes and zero-free regions for series-parallel graphs, with a resolution of pixels. Every orange pixel represent a provably zero-free value of , while every blue pixel represents a value of in the closure of the set of all chromatic zeroes of series-parallel graphs. The region depicted in the picture ranges from to .

We next restrict our attention to a subclass of series-parallel graphs. A leaf joined tree is a graph obtained from a rooted tree by identifying all its leaves except possibly into a single vertex. A while ago Sokal conjectured [12, Conjecture 9.5’] that for each integer the chromatic roots of all graphs all of whose vertices have degree at most except possibly one vertex are contained in the half plane . This conjecture was disproved by Royle for , as Sokal mentions in footnote 31 in [12]. Here we show that this is no coincidence, as we disprove this conjecture for all large enough.

Theorem 4.

There exists such that for all integers there exists a leaf joined tree obtained from a tree of maximum degree such that has a chromatic root with .

The proof of this theorem, together with some explicit calculations, also allows us to find such chromatic roots for . Table 1 in Section 6 records values of , which are accumulation points of chromatic zeroes of leaf joined trees, corresponding with the given .

1.1 Approach

Very roughly the main tool behind the proofs of our results is to write the chromatic polynomial as the sum of two other polynomials which can be iteratively computed for all series-parallel graphs, see Section 2 for the precise definitions. We also define the rational function and clearly implies . A certain converse also holds under some additional conditions.

To prove Theorem 1 we essentially show that these rational functions avoid the value . To prove presence of zeros we use that if the family rational functions behaves chaotically (in some precise sense defined in Section 4) near some parameter , then one can use the celebrated Montel theorem from complex analysis to conclude that there must be a nearby value and a graph for which

Our approach to obtaining density of chromatic zeros is similar in spirit to Sokal’s approach [11], but deviates from it in the use of Montel’s theorem. Sokal uses Montel’s ‘small’ theorem to prove the Beraha-Kahane-Weis theorem [2], which he is able to apply to the generalized theta graphs because their chromatic polynomials can be very explicitly described. It is not clear to what extent this applies to more complicated graphs. Our use of Montel’s theorem is however directly inspired by [6], which in turn builds on [9, 3, 4]. Our approach in fact also allows us to give a relatively short alternative proof for density of chromatic zeros of generalized theta graphs outside the disk , see Corollary 4.

Our proof of Theorem 4 makes use of an observation of Sokal and Royle in the appendix of the arXiv version of [10] (see https://arxiv.org/abs/1307.1721) saying that a particular recursion for ratios of leaf joined trees is up to a conjugation exactly the recursion for ratios of independence polynomial on trees. We make use of this observation to build on the framework of [6] allowing us to utilize some very recent work [1] giving an accurate description of the location of the zeros of the independence polynomial for the family of graphs with a given maximum degree.

Organization

The next section deals with formal definitions of series-parallel graphs and ratios. We also collect several basic properties there that are used in later sections. Section 3 is devoted to proving Theorem 1. In Section 4 we state a general theorem allowing us to derive various results on presence of chromatic zeros for series-parallel graphs. Finally in Section 5 we prove Theorem 4. We end the paper with some questions in Section 6

2 Recursion for ratios of series-parallel graphs

We start with some standard definitions needed to introduce, and setup some terminology for series-parallel graphs. We follow Royle and Sokal [10] in their use of notation.

Let and be two graphs with designated start- and endpoints , and respectively, referred to as two-terminal graphs. The parallel composition of and is the graph with designated start- and endpoints obtained from the disjoint union of and by identifying and into a single vertex and by identifying and into a single vertex The series composition of and is the graph with designated start- and endpoints obtained from the disjoint union of and by identifying and into a single vertex and by renaming to and to . Note that the order matters here. A two-terminal graph is called series-parallel if it can be obtained from a single edge using series and parallel compositions. From now on we will implicitly assume the presence of the start- and endpoints when referring to a two-terminal graph . We denote by the collection of all series-parallel graphs and by the collection of all series-parallel graphs such that the vertices and are not connected by an edge.

Recall that for a positive integer and a graph we have

where denotes the Kronecker delta. For a positive integer and a two-terminal graph , we can thus write,

(1)

where collects those contribution where receive the same color and where collects those contribution where receive the distinct colors. Since is equal to where is obtained from by adding an edge between and , both these terms are polynomials in . Therefore (1) also holds for any .

We next collect some basic properties of , and under series and parallel compositions in the lemma below. They can for example also be found in [11].

Lemma 1.

Let and be two two-terminal graphs and let us denote by an edge. Then we have the following equalities:

  • ,

  • ,

  • ,

  • ,

  • .

An important tool in our analysis of absence/presence of complex zeros is the use of the ratio defined as

(2)

which we view as a rational function in . We note that in case contains an edge between and , the rational function is constantly equal to . We observe that if , then and the converse holds provided .

The next lemma provides a certain strengthening of this observation for series-parallel graphs.

Lemma 2.

Let Then the following are equivalent

  • for some ,

  • for some ,

  • for some .

Proof.

We start with ‘(i) (ii)’. Let be as in the statement of the lemma such that for some series-parallel graph . Take such a graph with as few edges as possible.

By the above we way assume that , for otherwise (and hence ). Then also .

Suppose first that are not connected by an edge. By minimality, must be the parallel composition of two series-parallel graphs and such that, say and is not -connected, or in other words such that is a series composition of two smaller series-parallel graphs and . If we now identify vertices and of we obtain a series-parallel graph as the parallel composition of and (where vertices and have their roles reversed) for which . This is a contradiction since has fewer edges than . We conclude that in this case.

Suppose next that and are connected by an edge. We shall show that we can find another series-parallel graph , that is isomorphic to as a graph (and hence has as zero of its chromatic polynomial) but not as two-terminal graph. By the argument above we then have .

Let be obtained from by removing the edge . Then . If , then , contradicting the choice of . Therefore . If is a parallel composition of and , then , so there is a smaller graph, (namely or ), where is a zero, contradicting our choice of . Hence is the series composition of two graphs and . The graphs and cannot both be single edges, for otherwise would be a triangle. So let us assume that is not a single edge. We will now construct in a different way as series-parallel graph. First switch the roles of and in and denote the resulting series-parallel graph by . Then put in series with a single edge, and then put this in parallel with . In formulas this reads as . The resulting graph is then isomorphic to (but not equal to as a two-terminal graph). In case is not contained in , then is also not in . In that case we have

where is obtained from by first taking a series composition with an edge and then a parallel composition with an edge, that is, . This follows from the last item of Lemma 1, since for any two-terminal graph we have . So must be a zero of , or of . Because is not an edge, both and contain fewer edges than contradicting the choice of . Hence we conclude that is contained in , finishing the proof of the first implication.

The implication ‘(ii) (iii)’ is obvious. So it remains to show ‘(iii) (i)’.

To this end suppose that for some series-parallel graph . If the ratio equals , then clearly . So let us assume that the ratio equals . Then . Let us take such a graph with the smallest number of edges. By minimality, cannot arise as the parallel composition of two series-parallel graphs and by Lemma 1. Therefore must be equal to the series composition of two series-parallel graphs and . Now, as in the proof of ‘(i) (ii)’, identify vertices and of to form a new series-parallel graph , such that .

Let us finally consider the case that the ratio is equal to . In this case . If we now add an edge to connecting vertices and , creating the graph , then we have and we are done. ∎

We next provide a description of the behavior of the ratios behave under the series and parallel compositions. To simplify the calculations, we will look at the value of the modified ratio , which we call the effective edge interaction. Given define

(3)

the set of all values of the effective edge interaction at for the family of series-parallel graphs as a subset of the Riemann sphere, .

For any define the following Möbius transformation

and note that is an involution.

The next lemma captures the behavior of the effective edge interactions under series and parallel compositions and can be easily derived from Lemma 1.

Lemma 3.

Let be two two-terminal graphs with effective edge interactions respectively. Denote and for the effective edge interactions of the series and parallel composition of and respectively. Then

3 Absence of zeros near

In this section we prove Theorem 1. In the proof we will use the following condition that guarantees absence of zeros and check this condition in three different regimes. We first need a few quick definitions.

For a set , denote For subsets of the complex plane, we use the notation (and say is strictly contained in ) to say that the closure of is contained in the interior of . For we define to be the closed disk of radius centered at .

Lemma 4.

Let and let be a set satisfying: , , and . Then for all series-parallel graphs .

Proof.

By Lemma 2 it suffices to show that the ratios avoid the point . This corresponds to an effective edge interactions of since .

We first claim that all effective edge interactions are contained in , that is,

(4)

We show this by induction on the number of edges. The base case follows since . Assume next that and suppose that is the effective edge interaction of some series-parallel graph . If is the series composition of two series-parallel graphs and with effective edge interactions and respectively, then, by induction, and by assumption . If is the series composition of two series-parallel graphs and with effective edge interactions and respectively, then, by induction, and by assumption, , thereby proving (4).

It now suffices to show that . Suppose to the contrary that is the effective edge interaction some series-parallel graph . Take such a graph with as few edges as possible. If is the parallel composition of series-parallel graphs and with effective edge interactions and respectively, then , contradicting that If is the series composition of series-parallel graphs and with effective edge interactions and respectively, then . Thus and hence, say . But then , contradicting the choice of . This finishes the proof. ∎

Below we prove three lemmas allowing us to apply the previous lemma to different parts of the interval . First we collect two useful tools.

Lemma 5.

Let , then the circle with diameter is fixed by .

Proof.

First note that maps the real line to itself, because is real. Now let be the circle with diameter , this intersects the real line at right angles. The Möbius transformation sends to a circle through , and because is conformal the image must again intersect the real line at right angles. Therefore . ∎

Proposition 1.

Let be a disk. Then

Proof.

Obviously the second is contained in the first. The other inclusion is an immediate consequence of the Grace-Walsh-Szegő theorem. ∎

Now we can get into the three lemmas mentioned.

Lemma 6.

For each there exists a closed disk strictly contained in , satisfying , and .

Proof.

Let and choose real numbers with . They exist because and . Let be the closed disk with diameter . Clearly and . From Lemma 5 it follows that the boundary of is mapped to itself. Further, the interior point is mapped to which is also an interior point of . Therefore . Last, we see that , confirming all properties of . ∎

Lemma 7.

For each there exists a closed disk strictly contained in satisfying , and .

Proof.

The equation has a solution in , since and . Denote one such solution as . Then we see that and

Therefore there exists such that and . Let now be the closed disk with diameter and . By Lemma 5 we then know that . By construction we have that

and so satisfies the desired properties. ∎

Lemma 8.

There exists an open neighborhood around such that for each there exists a disk , satisfying , , and .

Proof.

Let . We claim that if is sufficiently small, there exists an such that satisfies the required conditions. Trivially, and , so we only need to show that , or equivalently .

We start with bounding the image of the disk :

So if we define , then . Since is an involution, we have

Now we claim that if is sufficiently small, then there exists such that . This is sufficient since for this value of we have

as desired.

We now prove the claim. As , the inequality is equivalent to

If we have a solution, then the quadratic polynomial in the variable should have real solutions, since its main coefficient is positive. Since the linear term is negative and the constant term is positive, both roots are positive. Thus it is sufficient to prove that the “smaller” real root is less then , i.e.

This is the case if

This means that we can take our set to be a disk of radius centered at . ∎

Now we are ready to prove Theorem 1.

Proof of Theorem 1.

For every we will now find an open around , such that does not contain chromatic roots of series-parallel graphs. For this follows directly from Lemmas 8 and 4. For and we appeal to Lemmas 6 and 7 respectively to obtain a closed disk with , and . We then claim that there is an open around , for which this disk still satisfies the requirements of Lemma 4 for all .
Certainly and remain true. Because holds, we can take small enough such that still holds, which confirms . Lastly, we know that . Because is compact, and the function depends continuously on , the inclusion remains true on a small enough open around . ∎

4 Activity and zeros

In this section we prove Theorems 2 and 3. We start with a theorem that gives a concrete condition to check for presence of chromatic zeros. For any we call any a virtual interaction.

Theorem 5.

Let If there exists either an effective edge interaction or a virtual interaction such that , then there exist arbitrarily close to and such that .

We will provide a proof for this result in the next subsection. First we consider some corollaries.

The first corollary recovers a version of Sokal’s result [11].

Corollary 1.

Let such that . Then there exists arbitrarily close to and such that .

Proof.

First of all note that . And therefore we have a virtual activity such that The result now directly follows from Theorem 5. ∎

Remark 1.

Recall that a generalized theta graph is the parallel composition of a number of equal length paths. Sokal [11] in fact showed that we can take in the corollary above to be a generalized theta graph. Our proof of Theorem 5 in fact also gives this. We will elaborate on this in Corollary 4 after giving the proof.

Our second corollary gives us Theorem 2.

Corollary 2.

Let . Then there exists arbitrarily close to and such that .

Proof.

Consider the map . We claim that for any . As , it is sufficient to show that for any . Or equivalently,

The maximal value of on the interval is (that is achieved at ), thus the claim holds.

To finish the proof, we choose such that . The result now follows from Theorem 5, since is an element of . ∎

Our next corollary gives us Theorem 3.

Corollary 3.

Let such that . Then there exists arbitrarily close to and such that .

Proof.

Consider the path of length . Its effective edge interaction is given by

Now the Möbius transformation maps the half plane to the complement of the unit disk, since , and the angle that the image of makes with at is degrees and since . The result now directly follows from Theorem 5. ∎

4.1 Proof of Theorem 5

We first introduce some definitions inspired by [6]. Let be a family of two-terminal graphs. Let . Then we call passive for if there exists an open neighborhood around such that the family of ratios is a normal family on , that is if any infinite sequence of ratios contains a subsequence that converges uniformly on compact subsets of to a holomorphic function . We call active for is is not passive for . We define the activity locus of by

(5)

Note that the activity locus is a closed subset of .

We next state Montel’s theorem, see [5, 8] for proofs and further background.

Theorem 6 (Montel).

Let be a family of rational functions on an open set . If there exists three distinct points such that for all and all , , then is a normal family on .

Montel’s theorem combined with activity and Lemma 2 give us a very quick way to demonstrate the presence of chromatic zeros.

Lemma 9.

Let and suppose that is contained in the activity locus of . Then there exists arbitrarily close to and such that .

Proof.

Suppose not. Then by Lemma 2, there must be an open neighborhood of on which family of ratios must avoid the points . Montel’s theorem then gives that the family of ratios must be normal on this neighborhood, contradicting the assumptions of the lemma. ∎

Lemma 10.

Let , and assume there exists an effective edge interaction or a virtual interaction such that . Then is contained in the activity locus of .

Proof.

We will first assume that , and for every open around find a family of series-parallel graphs such that is non-normal.

Let for some series-parallel graph . The virtual interaction is not a constant function of , because at the virtual interaction is . Therefore any open neighborhood of is mapped to an open neighborhood of and we may assume that is small enough, such that lies completely outside the closed unit disk. Now the pointwise powers converge to and the complex argument of the powers cover the entire unit circle for large enough.

Let us denote the unit circle by . Then is a straight line through for every . Inside the Riemann sphere, , these lines are circles passing through . For small enough and , and in a neighborhood of , these circles will lie in two sectors. More precisely, there exists large enough such that the argument of the complex numbers in are contained in two small intervals. Therefore we can find two sectors and around such that lies inside for all and lies outside of for all . Because the pointwise powers converge towards and the argument of the complex numbers are spread over the entire unit circle, there must be an for which intersects with both and . Then has points inside and outside the unit circle. Now the family is non-normal on . Indeed, the values inside the unit circle converge to , and the values outside the unit circle converge to . So any limit function of any subsequence can therefore not be holomorphic.

For the case with , we note again that this interaction cannot be a constant function of , because at the value must be 1. If we perform the same argument as above, we obtain a non-normal family of virtual interactions on . Applying to this family, produces a non-normal family of effective edge interactions on .

In both cases, we can conclude that is in the activity locus . ∎

Remark 2.

From the proof, we can extract the family of graphs which provides the non-normal family of interactions/ratios. In the case that we have a virtual interaction for a graph , the family consists of copies of in series, and copies of this in series.

For the case of an effective edge interaction , we instead put copies of in parallel, and copies of this in series.

Proof of Theorem 5.

For where either the interaction or the virtual interaction escapes the unit disk, the theorem is a direct consequence of Lemmas 9 and 10. If for there is an interaction or virtual interaction escaping the unit disk, this holds for all in a neighborhood as well. At these values, we already know that zeros accumulate, so they will accumulate at as well. ∎

We now explain how to strengthen Corollary 1 to generalized theta graphs. Let denote the family of all generalized theta graphs.

Corollary 4.

Let such that . Then there exists arbitrarily close to and such that .

Proof.

Note that is a virtual activity such that From Lemma 10 and Remark 2 we in fact find that is in the activity locus of . By Montel’s theorem we may thus assume that there exists such that . We claim that the ratio must in fact equal , meaning that is in fact a zero of the chromatic polynomial of the generalized theta graph .

The argument follows the proof of ‘(iii) (i)’ in Lemma 2. Suppose that the ratio is . Then we add an edge between the two terminals and realize that the resulting graph is equal to a number cycles glued together on an edge. Since chromatic zeros of cycles are all contained in , this implies that the ratio could not have been equal to . If the ratio equals , then we again obtain a chromatic zero of a cycle after identifying the start and terminal vertices. This proves the claim and hence finishes the proof. ∎

5 Chromatic zeros of leaf joined trees from independence zeros

This section is devoted to proving Theorem 4. Fix a positive integer and write . Given a rooted tree consider the two-terminal graph obtained from by identifying all leaves (except ) into a single vertex We take as the start vertex and as the terminal vertex of . Following Royle and Sokal [10], we call a leaf joined tree. We abuse notation and say that a leaf joined tree has maximum degree if all but its terminal vertices have degree at most We denote by the collection of leaf joined trees of maximum degree at most for which the start vertex has degree at most .

Our strategy will be to use Lemma 2 in combination with an application of Montel’s theorem, much like in the previous section. To do so we make use of an observation of Royle and Sokal in the appendix of the arXiv version of [10] saying that ratios of leaf joined trees, where the the underlying tree is a Cayley tree, are essentially the occupation ratios (in terms of the independence polynomial) of the Cayley tree. We extend this relation here to all leaf-joined trees and make use of a recent description of the zeros of the independence polynomial on bounded degree graphs of large degree due to the first author, Buys and Peters [1].

5.1 Ratios and occupation ratios

For a graph the independence polynomial in the variable is defined as

(6)

where the sum ranges over all sets of . (Recall that a set of vertices is called independent if no two vertices in form an edge of .) We define the occupation ratio of at as the rational function

(7)

where (resp. ) denotes the graph obtained from by removing (resp. and all its neighbors). We define for a positive integer , to be the collection of rooted graphs of maximum degree at most such that the root vertex, , has degree at most . We next define the relevant collection of occupation ratios,

A parameter is called active for if the family is not normal at .

We will use the following alternative description of . Define

and let be the family of rational maps, parametrized by , and defined by

  • the identify map is contained in

  • if , then .

Lemma 11 (Lemma 2.4 in [1]).

Let be an integer and write . Then