Explicit 3-colorings for exponential graphs

08/27/2018 ∙ by Adrien Argento, et al. ∙ 0

Let H=(V,E) denote a simple, undirected graph. The 3-coloring exponential graph on H is the graph whose vertex set corresponds to all (not necessarily proper) 3-colorings of H. We denote this graph by K_3^H. Two vertices of K_3^H, corresponding to colorings f and g of H, are connected by an edge in K_3^H if f(i) ≠ g(j) for all ij ∈ E. El-Zahar and Sauer showed that when H is 4-chromatic, K_3^H is 3-chromatic el1985chromatic. Based on this work, Tardif gave an algorithm to (properly) 3-color K_3^H whose complexity is polynomial in the size of K_3^H tardifAlg. Tardif then asked if there is an algorithm in which the complexity of assigning a color to a vertex of K_3^H is polynomial in the size of H. We present such an algorithm, answering Tardif's question affirmatively.

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1 Introduction

For a graph , we use and to denote its vertex and edge sets, respectively, and we use to denote its chromatic number. A homomorphism from a graph to a graph is a function from to such that for every edge in , is an edge in . We denote by the existence of a homomorphism from to . Note that a graph admits a proper -coloring if and only if .

The categorical product of two graphs has vertex set and edge set for and belonging to and , respectively. Observe that admits a homomorphism to both and . Since a proper colouring corresponds to a homomorphism to a complete graph, and since and , it is therefore immediate that . The following conjecture is due to Hedetniemi [Hed66] and was also posed as a question by Greenwell and Lovász [GL74].

Conjecture 1.1.

.

For two graphs and , there exists an exponential graph with the following property: only if . The vertices of are functions from into , and two functions are adjacent if for every edge of , is an edge of . With this definition of an exponential graph, observe that Hedetniemi’s conjecture can be rewritten: If and , then . But now we see that it is sufficient to replace with , and Conjecture 1.1 can be restated as follows.

Conjecture 1.2.

If , then .

The connection between exponential graphs and Hedetniemi’s conjecture was observed by El-Zahar and Sauer who proved Conjecture 1.2 when  [ES85]. (Exponential graphs have also been studied in other contexts [Lov67].) Specifically, El-Zahar and Sauer proved that if , then is

-colorable. Their proof is based on a global parity argument concerning so-called fixed points of odd cycles and does not immediately yield a 3-coloring of

. Attributing the question of efficiently finding a 3-coloring to Edmonds, Tardif presented an algorithm, implicit in the work of El-Zahar and Sauer, for 3-coloring  [Tar06]. Basically, Tardif noted that if we consider an (arbitrary) edge in any odd cycle of , then functions in which form a hitting set of the odd cycles in . A 3-coloring can easily be found based on a bipartition of the remaining vertices in . The time complexity of this algorithm depends on the time to find a bipartition, which is polynomial in the size of , but can be exponential in . A thorough description of this algorithm is provided in Section 1.2.

Tardif then posed the problem of finding an “explicit” 3-coloring of . Essentially, given a function belonging to a 3-chromatic component of , can we assign a color to this vertex in time polynomial in the size of ? (His precise question is a bit more involved and presented in detail in Section 1.2.) In this paper, we present an algorithm whose time complexity is linear in for finding such an explicit 3-coloring.

1.1 From coloring to coloring

Let be a positive integer, and let denote the odd cycle on the vertices . The vertex set consists of all functions from into . A vertex is a fixed point in if , where indices are computed modulo . Let denote the subgraph of induced on the following vertex set.

The problem of finding a 3-coloring of when can be reduced to the problem of finding a 3-coloring for  [ES85, Tar06]. Any non-isolated function from into contains an odd cycle with an even number of fixed points and such an odd cycle can be found in time polynomial in the size of (see Proposition 4.1 in [ES85] or Claim 2 from [Zhu98]).111Observe that the proof of this claim becomes easier when . Consider any function from into and partition into the set of vertices colored by and and the set of vertices colored by . Either the first set or the second set contains an odd cycle (which then has an even number of fixed points because it has at most two colors) or each set is bipartite, and we can color with four colors, which is a contradiction. For the connected component of containing this function , let us fix this odd cycle in to be . Now we find a 3-coloring for . Each function in the same connected component of as must also have an even number of fixed points on (see Lemma 3.3 in [ES85] or Claim 1 from [Zhu98]). So for each function in the same connected component as , the restriction of onto belongs to . Therefore, the vertex in corresponding to the function can be assigned the same color that the restriction of onto receives in the 3-coloring of .

1.2 Algorithm for 3-coloring

Now we are ready to present the algorithm for 3-coloring from [Tar06]. Let denote a fixed (but arbitrarily chosen) edge in .

Color-Graph() For all : If , assign the vertex in the color . Remove all the colored vertices from . Find a bipartition of the subgraph induced on the remaining vertices. For each vertex in , assign the vertex color . For each vertex in , assign the vertex color .

The correctness of this algorithm follows from the fact that the copies of in which form a hitting set for the odd cycles in , which follows from the main result of El-Zahar and Sauer (e.g., see Lemma 3.1 and Proposition 3.4 in [ES85]).

1.3 Tardif’s open problem

Tardif asked if there is an algorithm, whose running time is polynomial in , to assign a color to so that a 3-coloring is maintained for any subset of colored vertices of . (See Problem 6 in [Tar06] and also [Tar11].) He defines the vertex set

for a fixed (but arbitrarily chosen) edge . is a bipartite subgraph of induced on . If we can decide in time to which side of the bipartition belongs, then we can resolve Tardif’s question affirmatively. We present an algorithm for this task in the next section. Our approach is inspired by ideas from reconfiguration of 3-recolorings [CvdHJ07].

We note that Tardif showed that the main result from [ES85] implies that is bipartite. Conversely, our algorithm gives another proof that is bipartite, and consequently, we give an alternative proof of the main result of [ES85]. Note, however, that our proof is not completely independent as it uses Lemma 3.3 and Proposition 4.1 from [ES85].

2 Properties of adjacent functions

In this section, we state and prove two properties of functions that are adjacent in . These properties are key to the design and analysis of our algorithm, which we present in Section 3

. For an ordered pair of vertices

(i.e., an arc ) with colors and , respectively, we say the value of is , where

Monochromatic pairs have value 0.

Let denote a function from into . For a vertex , its color in is denoted by . We fix the orientation for the chords of length two in the cycle so that they form the directed cycle , which we refer to as . Then we have the following definitions. Recall that is a fixed (but arbitrarily chosen) edge in .

Definition 2.1.

The label of , denoted by , is the total value of the arcs in based on . Formally,

Definition 2.2.

The little path of , denoted by , is the directed path from to in containing arcs (e.g., see Figure 1). The value of , denoted by , is computed as follows.

Figure 1: The dotted edges denote . The directed edges denote . The little path is shown in red.
Observation 2.3.

.

Observation 2.4.

If , then .

Now let and be two functions from into such that and are adjacent. Recall that denotes the vertex set . Let denote the copy of in and let denote the copy of in . Define the directed cycle as follows.

(1)

We have

(2)

where subscripts are computed modulo . We can relate the value of the arcs in to the value of the arcs in using the following claim.

Claim 2.5.

Let and be two functions from into such that and are adjacent. Then

Proof..

Since and are adjacent, we have and for . If , then . Furthermore, we have the following observations.

  1. , and

  2. .

Next, we present two key properties of adjacent functions via the following lemmas.

Lemma 2.6.

Let and be two functions from into such that and are adjacent. Then .

Proof.

Recall that is defined as follows.

By Claim 2.5, we observe that . The same argument shows that .∎

Lemma 2.7.

Let and be two functions from into such that and are adjacent and let . Then .

Proof.

Without loss of generality, let and . Then

and

Applying Claim 2.5, we have

Since

the lemma follows. ∎

3 Coloring a vertex of in time

Now we present an algorithm to assign a color to a function when (i.e., and ). Our algorithm will produce a 3-coloring for and the time required to assign a color to is . We recall that edge is a fixed edge and is considered as input to the algorithm.

Color-Vertex() If , assign color . Otherwise, compute value . If , assign color . If , assign color .

The correctness of the algorithm will be shown via the following theorem, whose proof is based on the observation that if and are adjacent in , then by Lemma 2.7, and are anticorrelated. For example, if , then either is positive and is negative or vice versa. Theorem 3.1 implies that using Color-Vertex to color the vertices of results in a proper 3-coloring of .

Theorem 3.1.

Let and be two functions from such that and are adjacent. Then Color-Vertex assigns and different colors.

First, we show via Claim 3.2 that Step 2. of Color-Vertex is well-defined.

Claim 3.2.

Let be a function in . Then either or .

Proof..

By Observation 2.3, is a multiple of 3. Since , it follows that is a multiple of 2. Thus, is an integer and is a multiple of 3.

By Observation 2.4 and the fact that , is not a multiple of 3. Therefore, . Thus, we have

Next, we show that two adjacent functions in will not be assigned to the same side of the bipartition, and thus will not be assigned the same color.

Claim 3.3.

Let and be two adjacent functions in and let . Then and cannot both be greater (or both smaller) than .

Proof..

By Lemma 2.7, we have

Let us consider two cases. The first case is when . Then we have

By Claim 3.2, we conclude that . The second case is when . Then we have

By Claim 3.2, we conclude that .

4 Explicit homomorphisms for odd cycles

For a graph and a cycle for odd integer , one can define (as in the introduction) the exponential graph . The vertices of are functions from into and two such functions and are adjacent if for all edges of , and are adjacent in . Häggkvist, Hell, Miller and Neumann Lara proved that if there is no homomorphism from to , then has a homomorphism to  [HHML88]. Their proof can be viewed as a generalization of the work of El-Zahar and Sauer, who proved the same statement when . In fact, as in the case in the latter proof of El-Zahar and Sauer, it is also implicit in the proof of Häggkvist et al. that if we consider an (arbitrary) edge in any odd cycle of , then functions in which form a hitting set of the odd cycles in . It is therefore not surprising that we can extend our framework for obtaining explicit homomorphisms to odd cycles.

As in the case of , we can find a homomorphism from to by considering an arbitrary odd cycle in . Applying Lemma 7 from [HHML88], we see that if there is no homomorphism from to , then contains an odd cycle with an even number of fixed points.222In [HHML88], a fixed point is called a 2-point. We refer to this odd cycle as . It remains to generalize the two key properties of adjacent functions (i.e., Lemmas 2.6 and 2.7). For an ordered pair of vertices (i.e., an arc ) with values and (from ), respectively, we say the value of is , where

(3)

For example, we have

Let and be two adjacent functions from to . We can apply the rules from (3) to compute the values and via Definitions 2.1 and 2.2. Note that if for some arc , then is an isolated function in . Moreover, note that if for some arc , then and are not adjacent.

The next lemmas are the generalizations of Lemma 2.6 and Lemma 2.7 for homomorphisms to an odd cycle .

Lemma 4.1.

Let and be two functions from into such that and are adjacent. Then .

Proof.

Recall the definition of the directed cycle and from (1) and (2). It is straightforward to prove that . ∎

Lemma 4.2.

Let and be two functions from into such that and are adjacent and let . Then .

Proof.

Let and . We observe that

Since

the lemma follows. ∎

It is straightforward to extend Claims 3.2 and 3.3 to this generalized setting and we obtain the following theorem.

Theorem 4.3.

Let and be two functions from into such that and are adjacent. Then Color-Vertex assigns and to adjacent vertices in .

5 Discussion: Explicit versus efficient colorings

For a graph such that , the question of finding an explicit 3-coloring of is closely related to—but not exactly the same as—the question of finding an efficient 3-coloring of . A connected component of is either (i) isolated (i.e., a single vertex), (ii) bipartite, or (iii) 3-chromatic. For a function from into , it can be efficiently determined (in time polynomial in the size of ) whether or not is isolated. For any given connected component of that is bipartite or 3-chromatic, there exists an odd cycle that can be found efficiently (as discussed in Section 1.1) and this odd cycle can be used to obtain an explicit and efficient 3-coloring for this component. In other words, for a given connected component, after a polynomial amount of preprocessing time (i.e., time to find an odd cycle and to fix an orientation of its chord cycle and an edge to use as input for the Vertex-Color routine), we give an explicit reason (i.e., certificate) for assigning a particular color to a function in the given connected component. In particular, the value of the little path from to is such a short certificate. In terms of efficiency, for any subset of functions in belonging to a fixed connected component, the total time required to color the subgraph induced on is .

Moreover, for a function belonging to any 3-chromatic component of , we can actually use an arbitrary fixed cycle from for the Color-Vertex routine. However, the fact that we can use the same cycle in for any such follows from the main result of El-Zahar and Sauer; Proposition 3.4 in [ES85] states that for such an , all odd cycles in have an even number of fixed points. Note that the results we have presented here do not imply a proof of this proposition. Thus, while such a function can in fact be assigned a color efficiently (i.e., in time ), this time complexity is not implied solely by the results we have presented here.

For belonging to an arbitrary bipartite component of , using the approach presented in this paper, we can assign a color to in time , where is the set of functions for which we searched for a new odd cycle containing an even number of fixed points. Note that is upper bounded by the number of functions also belonging to bipartite components previously colored by the algorithm. In other words, the algorithm is input-sensitive; before invoking the Color-Vertex routine on , we need to check all odd cycles used so far (in the order used) until we find one with an even number of fixed points with respect to . We leave it as an open problem to find an algorithm that assigns a color in time to a vertex from a bipartite component of , so that the resulting coloring is a proper 3-coloring or possibly even a 2-coloring. Finally, we note that we do not know how to efficiently determine if a function belongs to a bipartite component (i.e., whether or not it contains at least one odd cycle with an odd number of fixed points).

6 Acknowledgements

We thank Stéphan Thomassé for telling us about Hedetniemi’s Conjecture and for, even earlier, telling us about the results in [CvdHJ07].

References

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