Low Degree Nullstellensatz Certificates for 3-Colorability

In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most $12$ vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when $\mathbb{k}=GF(2)$.


Introduction
In recent years, combinatorial optimization has flourished from algorithms that fundamentally rely on tools from algebraic geometry and commutative algebra. Work of Lasserre [9], Lovász-Schrijver [11], Sherali-Adams [13], Gouveia, Parrilo and Thomas [7], and many others have used polynomials to develop approximation algorithms for optimization problems. Another recent algorithm akin to those above is the Nullstellensatz Linear Algebra algorithm (NulLA) of De Loera et. al [4] which addresses feasibility issues in polynomial optimization. Given a set of polynomials f 1 , f 2 , . . . , f s ∈ [x 1 , . . . , x n ] for some field , NulLA's goal is to certify that the system of equations f 1 = 0, f 2 = 0, . . . f s = 0 has no solution in , the algebraic closure of . It exploits Hilbert's Nullstellensatz, a celebrated and fundamental theorem in algebraic geometry (see [2]). The polynomials α 1 , α 2 , . . . , α s are referred to as a Nullstellensatz certificate of infeasibility; indeed they are a witness that the polynomial system f 1 = 0, f 2 = 0, . . . , f s = 0 has no solution. The maximum degree of the α ′ i s is referred to as the Nullstellensatz degree of the system, and it is a measure of the complexity of certifying that the system of polynomial equations has no solutions. If a system of polynomial equations is known to have a Nullstellensatz certificate whose Nullstellensatz degree is a small constant (and if is finite), one can find a Nullstellensatz certificate in polynomial time in the number of variables through a sequence of linear algebra computations (see [4] for details). However, for general polynomial systems, it is well known that the degree of Nullstellensatz certificates can grow as a function of the number of variables.
The underlying paradigm in all the above algorithms is the construction of iterative approximations that are tractably computable at early stages. When applied to combinatorial optimization problems, particularly graph theoretic ones, the key problem that arises is determining the classes of graphs for which a given problem can be resolved in early iterations. For instance, when applied to the stable set problem, Gouveia, Parrilo and Thomas [7] show that the first iteration of the theta body hierarchy solves the stable set problem for perfect graphs. This result was first established by Lovász [10] by exploiting polynomials as well. NulLA itself was originally introduced as a means of unfolding classes of non-3-colorable graphs that can be detected to be non-3-colorable efficiently (that is, in polynomial time in the number of variables of a given graph). In particular, the authors of [4] applied the NulLA algorithm to the following algebraic formulation of graph 3-colorability due to Bayer. We will refer to this as Bayer's formulation throughout the manuscript. Lemma 1.2 (Bayer [1]). A graph G with vertex set V and edge set E is 3-colorable if and only if the following system of equations has a solution over an algebraically closed field with char( ) = 3.
The fundamental concern then is determining combinatorial features of non-3-colorable graphs that dictate the minimum Nullstellensatz degree of infeasibility for the system in Lemma 1.2, which we denote by N (G), is a small constant. In light of this, it is natural to address the following problem, a variant of which was first asked in [3]: Problem 1.3. Given a finite field , and positive integer d, characterize those graphs with N (G) = d.
Computational evidence (see, for example, Table 1 of [6]) suggests that the minimum Nullstellensatz degree of a non-3-colorable under Bayer's formulation is smallest when the field of coefficients chosen is GF (2) as opposed to GF (p) for primes p > 2, so from a computational complexity perspective, it may be beneficial to begin addressing Problem 1.3 by working with Bayer's formulation when = GF (2). A partial answer in this case was given by De Loera et. al [3] (see their paper for relevant definitions). Theorem 1.4 (Theorem 2.1 of [3]). For a given simple undirected graph G with vertex set V = {v 1 , v 2 , . . . , v n } and edge set E, the polynomial system over GF (2) encoding the 3-colorability of G E} has a degree one Nullstellensatz certificate of infeasibility if and only if there exists a set C of oriented partial 3-cycles and oriented chordless 4-cycles from Arcs(G) such that where C (v i ,v j ) denotes the set of cycles in C in which the arc (v i , v j ) ∈ Arcs(G) appears.
This characterization adds directed structure to undirected graphs, and hence does not fully capture an inherent combinatorial characterization directly from the graphs themselves. In this paper, we provide such a direct combinatorial characterization for Bayer's formulation when = GF (2). Before introducing the combinatorial characterization, we define the following class of graphs that will play a key role.
. . , v n } and edge set E is covered by length 2 paths if there exists a set C of length 2 paths in G such that (1) each edge in E appears in an even number of paths in C, then the number of paths in C with v i and v j as endpoints is even.
We now present the main combinatorial characterization.   Example 1.7. Let n be a positive integer. The graph W n , referred to as the wheel graph, is the graph whose vertex set is {v 1 , v 2 , . . . , v n , w} where the induced subgraph on V ′ = {v 1 , v 2 , . . . , v n } is a cycle and w is a vertex adjacent to all vertices in V ′ . Without loss of generality, we may assume the cycle whose vertex set is V ′ has edge set {v 1 v 2 , . . . , v n−1 v n , v n v 1 }. When n is odd, W n is not 3-colorable. Observe that C = {v 1 wv 2 , v 2 wv 3 , . . . , v n−1 wv n , v n wv 1 } is a set of length 2 paths in W n that satisfies all the conditions in Definition 1.5. We conclude that NulLA (applied to Bayer's formulation when = GF (2)), detects that odd wheels are non-3-colorable with a degree 1 Nullstellensatz certificate. This suggests that NulLA has the potential to be used not only as a computational tool but as a tool for automatic theorem proving.
Example 1.8. Let G be the graph depicted in Figure 1. This non-3-colorable graph is known as the Moser spindle and is ubiquitous in graph theory. Using Theorem 1.6, we will show that N (G) > 1 when = GF (2). Suppose otherwise. By Theorem 1.6, G is covered by length 2 paths, which we refer to collectively as C.
Since the edge v 1 v 7 is not on a 3-cycle, v 1 v k is not an edge. Moreover, since the edge v 1 v 7 is not on a 4-cycle, v 1 v 7 v k is the only member of C whose endpoints are v 1 and v k . But this is impossible because v 1 v k is not an edge, so the number of paths in C with v 1 and v k as endpoints must be even. Hence, there are no length 2 paths in C with v 1 and v 7 as endpoints. Thus, through Theorem 1.6, C certifies that N (G\v 1 v 7 ) = 1. But this is impossible since G\v 1 v 7 is 3-colorable, by observation. Thus, no set C with the desired property exists, so N (G) = 1. Example 1.8 generalizes directly in the following way, providing a combinatorial obstruction to existence of a degree 1 Nullstellensatz certificate for Bayer's formulation when = GF (2). Corollary 1.9. Let = GF (2), and suppose G is a non-3-colorable graph that contains an edge e for which the following are true: • G\e is 3-colorable, and • e is not an edge in a 3-cycle nor a 4-cycle of G. Then N (G) > 1.
Remark 1.10. One of the most celebrated constructions of very hard instances of graph 3colorability is a construction of Mizuno and Nishihara [12]. Corollary 1.9 is consistent with their findings. Indeed, in all the graphs they present in Figure 3 (see [12]), the removal of any edge leaves a 3-colorable graph, and each of these graphs has an edge that does not lie on 3 or 4-cycle. This implies that when = GF (2), N (G) > 1 for such graphs G, so computationally determining that they are not 3-colorable is not immediate under the NulLA paradigm.
Alongside our combinatorial characterization, in Section 3 we begin the program of determining the Nullstellensatz degree of Bayer's formulation (with coefficients in GF (2)) for small non-3-colorable graphs. Most notably we prove Theorem 1.11. If = GF (2) and |V (G)|≤ 12, then N (G) ≤ 4.

Characterizing Degree 1 Certificates
This section is dedicated to proving Theorem 1.6, and in particular developing a combinatorial characterization of non-3-colorable graphs G for which N (G) = 1 (under Bayer's formulation with coefficients in GF (2)). We begin with a technical proposition that will be needed throughout: (1) N (G) = 1.
(2) 1 is a -linear combination of (3) 1 is a -linear combination of (4) 1 is a -linear combination of The above proposition finds alternate and equivalent sets of polynomials whose solution sets are the same as that of the system in Lemma 1.2. The last set of polynomials are particularly useful in uncovering our combinatorial characterization. The equivalence of the first three sets was proven in Theorem 2.1 of [3]. The equivalence to the last set of polynomials follows an argument similar to the proof of Theorem 2.1 in [3]. For completeness, we include a proof of this equivalence in the appendix.
In proving Theorem 1.6, we will repeatedly appeal to the following immediate proposition: Proposition 2.2. Let G be a graph with vertex set V = {v 1 , v 2 , . . . , v n } and edge set E, and suppose G is covered by a set C of length 2 paths. The following statements are equivalent: (1) The number of pairs v i , v j ∈ V with i < j for which there are an odd number of paths in C containing v i with v j as an endpoint, is itself an odd number. (2) The sum over all pairs i < j of the number of paths in C containing v i with v j as an endpoint is odd.
We now move on to the main result: Proof. (of Theorem 1.6) Throughout this proof, for any set of polynomials S in a polynomial ring whose coefficients are in , we denote by S the linear span of S over . Let F be the following set of polynomials: By Proposition 2.1, we know that N (G) = 1 if and only if 1 ∈ F , so we must show that 1 ∈ F if and only if G is covered by a set C of length 2 paths. First suppose G is covered by a set of length 2 paths C. Consider the set H ⊂ F consisting of the following polynomials: (1) j + x j x 2 i + 1 for each v i , v j ∈ V with i < j such that the number of length 2 paths in C containing v i and having v j as an endpoint is odd. We claim 1 ∈ H and hence 1 ∈ F . Observe that the non-constant monomials appearing in F (and hence in H) are all of the form x 2 r x s , where v r , v s ∈ V are arbitrary. We start by showing that the coefficient of x 2 r x s in h∈H h is 0 in , and so all non-constant terms in h∈H h vanish. An Thus, the coefficient of x 2 r x s in the combined contribution from the terms in (a),(b),(c),(d) is 0 in . Finally we need to discern the constant term of h∈H h. The constant term is the parity of the number of summands of the form (2). But this is immediately 1 by condition (2) of Definition 1.5 and Proposition 2.2, so the constant term in h∈H h over is 1.
We must now show that if N (G) = 1, then such a set C exists. In that light, Proposition 2.1 asserts the existence of a set H of polynomials of the form (9) and (10) with h∈H h = 1. Let H * be the restriction of H to the polynomials in (10). Construct C to consist of the paths v i v j v k for which Suppose v r v s ∈ E, define S r,s to be the sum (in ) of the coefficients of the monomials x 2 r x s and x r x 2 s appearing in h∈H * h. Since h∈H h = 1, and the only other summand of h∈H h not in h∈H * h is x 2 r x s + x r x 2 s + 1, S r,s is 0 in . However, the contribution of a single summand if v r and v s are endpoints of v i v j v k , and 0 otherwise. Since S r,s is 0 in , we deduce that the edge v r v s lies on an even number of paths in C.
If v r , v s ∈ V but v r v s ∈ E, then any x 2 r x s term in h∈H h appears in h∈H * h, so the coefficient of x 2 r x s in h∈H * h is 0 in . But x 2 r x s appears once in the summand of H * corresponding to the path v i v j v k precisely when v r and v s are endpoints of v i v j v k . Thus, the number of paths whose endpoints are v r and v s is even.
Each edge v i v j with i < j contributes a 1 to the sum h∈H h. Moreover, for such pairs i, j the monomial x 2 i x j appears an odd number of times in H * . We know x 2 i x j appears in H * once for each path in C whose endpoints are v i and v j and once for each path in C with v i as a midpoint and v j as an endpoint. This is equivalent to x 2 i x j appearing in H * once for each path in C containing v i with v j as an endpoint. Since the number of 1s appearing in h∈H h is odd, there are an odd number of i, j pairs with i < j such that the number of paths in C containing i with j as an endpoint is odd. By Proposition 2.2, this establishes condition (2) of Definition 1.5.

NulLA on Small Graphs and Future Directions
Section 2 equipped us with a complete combinatorial understanding of the graphs G for which N (G) = 1 when = GF (2). By Theorem 2.1 of [6], N (G) ≡ 1 (mod 3), so in investigating Problem 1.3, the next natural step is determining when N (G) = 4. This section is devoted to a systematic study of this for small graphs. We first remark that, in order to find graphs with low minimum Nullstellensatz degree, we only need to focus on a subclass of non-3-colorable graphs.  [14]). A non-3-colorable graph G is 4-critical if for any edge e ∈ E(G), G\e is 3-colorable.
The following observation is fundamental for our purposes. See Chapter 5 of [14] for a discussion of this. Lemma 3.2 ([14]). Every non-3-colorable graph has a 4-critical subgraph.
In light of the previous lemma, the following lemma tells us that 4-critical graphs provide upper bounds for the minimum Nullstellensatz degree of general non-3-colorable graphs. 4  1  0  1  5  0  0  0  6  1  0  1  7  1  1  2  8  2  3  5  9  5  16  21  10  13  137  150  11  38  1183  1221  12  141  14440  14581  Total  202  15780  15982  Table 1. N (G) for 4-critical graphs on at most 12 vertices when = GF (2). Lemma 3.3 (Lemma 3.14 of [5]). If H and G are non-3-colorable graphs with H a subgraph of G, then N (H) ≥ N (G). Lemma 3.2 and Lemma 3.3 suggest that we solely focus on minimum degree Nullstellensatz certificates for 4-critical graphs. We computed minimum degree Nullstellensatz certificates for all such graphs on at most 12 vertices. A summary of the results are illustrated in Table 1. With these observations we can now prove Theorem 1.11.

|V | N (G) = 1 N (G) = 4 Total 4-critical graphs
Proof. (of Theorem 1.11) By Lemma 3.2, every non-3-colorable graph G on at most 12 vertices has a 4-critical subgraph H. By Table 1 Many pertinent questions arise from our study of the minimum Nullstellensatz degree of graphs under Bayer's formulation. First and foremost, unless P=NP, one should expect to find a family of graphs for which the minimum Nullstellensatz degree grows arbitrarily large (see Lemma 3.2 of [5] for a discussion of this). As evidenced by Table 1, an exhaustive search of the almost 16,000 4-critical graphs on at most 12 vertices indicates that a first step in this direction is to address the following problem: Problem 3.4. For any positive integer t, find a graph G for which N (G) > t.
After exhaustive experimentation, the authors of [5] have yet to see an example resolving Problem 3.4 when t = 4 for any finite field . One possible method for addressing Problem 3.4 is understanding what happens to the minimum Nullstellensatz degree under the famous Hajós construction. In his seminal paper [8], Hajós defined a recursively constructed class of graphs, which he called 4-constructible, in the following way: i) K 4 is 4-constructible. ii) For any two non-adjacent vertices u and v in a 4-constructible graph G, the graph obtained from G by adding an edge e incident to u and v and contracting e is also 4-constructible. iii) (Hajós Construction) For any two 4-constructible graphs G and H, with vw an edge of G, and xy an edge of H, the graph obtained by identifying v and x, removing vw and xy, and adding the edge wy is also 4-constructible. Hajós proved that the set of 4-constructible graphs is precisely the set of 4-critical graphs, so it is fundamental for us to determine what changes in the minimum Nullstellensatz degree of graphs when applying these constructions. Observe N (K 4 ) = 1, and by Lemma 3.14 of [5], the minimum Nullstellensatz degree will not increase by applying construction ii). This leads us to the following fundamental question: Problem 3.5. Let G and H be 4-critical graphs. What is the relationship between N (G), N (H) and the minimum Nullstellensatz degree of the graph obtained from G and H by applying the Hajós construction?
Finally, suppose v i v k / ∈ E, v j v k ∈ E. This is the only remaining case since the case when v i v k ∈ E, v j v k / ∈ E follows by symmetry. In this case, the only polynomials in (5) and (6) that contain x i x j x k as a summand are Again, since the coefficient of x i x j x k must be 0, either neither of these appear as summands in (11) or both do. Again, we only need consider the case when both appear. In this case, again since = GF (2), β ijk = β kij = 1. Observe then that the contribution of such polynomials to the right hand side of (11) is i + x k x 2 j , and the latter polynomial is a polynomial in (8) since v i v j , v j v k ∈ E. Thus if 1 is a -linear combination of polynomials in (6) then it is a -linear combination of polynomials in (8).