Between 2- and 3-colorability

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n,\,1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$.


Introduction
The determination of the chromatic number of G n,p , where p = c n for constant c, is a central topic in the theory of random graphs. For 0 < c < 1, such graphs contain, in expectation, a bounded number of cycles, and are almost-surely 3-colorable. The chromatic number of such a graph may be 2 or 3 with positive probability, according as to whether or not any odd cycles appear.
For c ≥ 1, we find that the chromatic number χ(G n, c n ) ≥ 3 with high probability, and letting c k := sup c χ(G n, c n ) ≤ k, it is known for all k and c ∈ (c k , c k+1 ) that χ(G n, c n ) ∈ {k, k + 1}, see Luczak [7] and Achlioptas and Naor [2]; for k > 2, the chromatic number may well be concentrated on the single value k, see Friedgut [5] and Achlioptas and Friedgut [1].
In this paper, we consider finer notions of colorability for the graphs G n, c n for c ∈ (1, c 3 ), by considering homomorphisms from G n, c n to odd cycles C 2ℓ+1 . A homomorphism from a graph G to C 2ℓ+1 implies a homomorphism to C 2k+1 for k < ℓ. As the 3-colorability of a graph G corresponds to the existence of a homomorphism from G to K 3 , the existence of a homomorphism to C 2ℓ+1 implies 3-colorability. Thus considering homomorphisms to odd cycles C 2ℓ+1 gives a hierarchy of 3-colorable graphs amenable to increasingly stronger constraint satisfaction problems. Note that a fixed graph having a homomorphism to any odd-cycle is bipartite.
Our main result is the following: Theorem 1. For any ℓ > 1, there is an ε > 0 such that with high probability, G n, 1+ε n either has odd-girth < 2ℓ + 1 or has a homomorphism to C 2ℓ+1 .
Conversely, we expect the following: Conjecture 1. For any c > 1, there is an ℓ c such that with high probability, there is no homomorphism from G n, c n to C 2ℓ+1 for ℓ ≥ ℓ c .
As c 3 is known to be at least 4.03, the following confirms Conjecture 1 for a significant portion of the interval (1, c 3 ).
Theorem 2. For any c > 2.774, there is an ℓ c such that with high probability, there is no homomorphism from G n, c n to to C 2ℓ+1 for ℓ ≥ ℓ c .
We also have that ℓ 4 = 2: Theorem 3. With high probability, G n, 4 n has no homomorphism to C 5 .
Note that as c 3 > 4.03 > 4, we see that there are triangle-free 3-colorable random graphs without homomorphisms to C 5 . Our proof of Theorem 3 involves computer assisted numerical computations. The same calculations which rigorously demonstrate that ℓ 4 = 2 suggest actually that ℓ 3.75 = 2 as well.
Our results can be reformulated in terms of the circular chromatic number of a random graph.
Recall that the circular chromatic number χ c (G) of G is the infimum r of circumferences of circles C for which there is an assignment of open unit intervals of C to the vertices of G such that adjacent vertices are assigned disjoint intervals. (Note that if circles C of circumference r were replaced in this definition with line segments S of length r, then this would give the ordinary chromatic number χ(G).) It is known that is always rational, and moreover, that χ c (G) ≤ p q if and only if G has a homomorphism to the circulant graph C p,q with vertex set {0, 1, . . . , q − 1}, with v ∼ u whenever dist(v, u) := min{|v − u| , v + q − u, u + q − v} ≥ q. (See [9].) Since C 2ℓ+1,ℓ is the odd cycle C 2ℓ+1 our results can be restated as follows: Theorem 4. In the following, inequalities for the circular chromatic number hold with high probability.
Note that for any c and ℓ > 1, there is positive probability that G n, c n has odd girth < 2ℓ + 1, and a positive probability that it does not. In particular, as the probability that G n, c n has small odd-girth can be computed precisely, Theorem 1 gives an exact probability in (0, 1) that G n, 1+ε n has a homomorphism to C 2ℓ+1 . Indeed, Theorem 1 implies that if c = 1 + ε and ε is sufficiently small relative to ℓ, then where .
We close with two more conjectures. The first concerns a sort of pseudo-threshold for having a homomorphism to C 2ℓ+1 : Conjecture 2. For any ℓ, there is a c ℓ > 1 such that G n, c n has no homomorphism to C 2ℓ+1 for c > c ℓ , and has either odd-girth < 2ℓ + 1 or has a homomorphism to C 2ℓ+1 for c < c ℓ .
The second asserts that the circular chromatic numbers of random graphs should be dense.

Conjecture 3.
There are no real numbers 2 ≤ a < b with the property that for any value of c, Pr(χ c (G n, c n ) ∈ (a, b)) → 0.
Note that our Theorem 1 confirms this conjecture for the case a = 2.

Structure of the paper
We prove Theorem 1 in Section 3. We first prove some structural lemmas and then we show, given the properties in these lemmas, that we can algorithmically find a homomorphism. We prove Theorem 2 in Section 4 by the use of a simple first moment argument. We prove Theorem 3 in Section 5. This is again a first moment calculation, but it has required numerical assistance in its proof.
3 Finding homomorphisms Lemma 1. If α < 1/10 and c is a positive constant where then w.h.p. any two cycles of length less than α log n in G n,p , p = c n , are at distance more than α log n.

Proof
If there are two cycles contradicting the above claim, then there exists a set S of size s ≤ 3α log n that contains at least s + 1 edges. The expected number of such sets can be bounded as follows: ✷ Our next lemma is concerned with cycles in K 2 which is the 2-core of G n,p . The 2-core of a graph is the graph induced by the edges that are in at least one cycle. When c > 1, the 2-core consists of a linear size sub-graph together with a few vertex disjoint cycles. By few we mean that in expectation, there are O(1) vertices on these cycles.
The degree sequence of K 2 can be generated as follows, see for example Aronson, Frieze and Pittel [3]: Let λ be the solution to We deduce from this that We generate the degrees d(1), d(2), . . . , d(ν) as independent copies of the random variable Z where for d ≥ 2, .
We condition that the sum It is shown in [3] that if Z 1 , Z 2 , . . . , Z N are independent copies of Z then We observe next that the maximum degree in G n,p and hence in K 2 is q.s. 1 at most log n. It follows from this and (2) that Lemma 2. For any α, β, there exists c 0 > 1 such that w.h.p. any cycle of length greater than α log n in the 2-core of G n,p , p = c n , 1 < c < c 0 , has at most β log n vertices of degree ≥ 3.
We will show then that w.h.p. the K 2 does not contain a cycle C where (i) |C| ≥ α log n and (ii) C contains β|C| vertices of degree greater than two.
We can bound the probability of the existence of a "bad" cycle C as follows: In the following display we choose the vertices of our cycle in ν k ways and then arrange these vertices in a cycle C in (k − 1)!/2 ways. Then we choose βk vertices to have degree at least three. We then sum over possible degree sequences for the vertices in C. This explains the factor θ k k i=1 . We now resort to using the configuration model of Bollobás [4]. This would explain the product k 2µ−2i+1 . We use the denominator 2µ − k to simplify the calculation. The configuration model computation will inflate our estimate by a constant factor that we hide with the notation ≤ b . We write ✷ Lemma 3. For any α and any k ∈ N, there exists ε 0 > 0 such that w.h.p. we can decompose the edges of the G = G n,p , p = 1+ε n , 0 < ε < ε 0 , as F ∪ M, where F is a forest, and where the distance in F between any two edges in M is at least k.

Proof
By choosing β < 1 2k in Lemma 2 we can find, in every cycle of length > α log n of the 2-core K 2 of G (which includes all cycles of G), a path of length at least 2k + 1 whose interior vertices are all of degree 2. We can thus choose in each cycle of K 2 of length > α log n such a path of maximum length, and let P denote the set of such paths. (Note that, in general, there will be fewer paths in P than long cycles in K 2 due to duplicates, but that the elements of P are nevertheless disjoint paths in K 2 .) We now choose from each path in P an edge from the center of the path to give a set M 1 . Note that the set of cycles in G \ M 1 is the same as the set of cycles in G \ P ∈P P . (In particular, the only cycles which remain have length ≤ α log n and are at distance ≥ k from M.) Thus, letting M 2 consist of one edge from each cycle of G \ M 1 , Lemma 1 implies that M = M 1 ∪ M 2 is as desired. ✷ Proof of Theorem 1. Our goal in this section is to give a C 2ℓ+1 -coloring of G = G n, 1+ε n for ε > 0 sufficiently small. By this we will mean an assignment c : V (G) → {0, 1, . . . , 2ℓ} such that x ∼ y in G implies that c(x) ∼ c(y) as vertices of C 2ℓ+1 ; that is, that x = y ± 1 (mod 2ℓ + 1).
Consider a decomposition of G as F ∪ M as given by Lemma 3, with k = 4ℓ − 2.
We begin by 2-coloring F . Let c F : V → {0, 1} be such a coloring. Our goal will be to modify this coloring to give a good C 2ℓ+1 coloring of S.
Let B be the set of edges xy ∈ M for which c F (x) = c F (y), and let B be a set of distinct representatives for B, and for i = 0, 1, let We now define a new C 2ℓ+1 coloring c : V → {0, 1, . . . , 2ℓ}, by (Color addition and subtraction are computed modulo 2ℓ + 1.) Since edges in M are separated by distances ≥ 4ℓ − 2, this coloring is well-defined (i.e., there is at most one choice for x). Moreover, c is certainly a good C 2ℓ+1 -coloring of F . Thus if c is a not a good C 2ℓ+1 -coloring of S, it is bad along some edge xy ∈ M. But if such an edge was already properly colored in the 2-coloring c F , it is still properly colored by c, since it has distance ≥ 4ℓ − 2 ≥ 2ℓ − 1 from other edges in M. On the other hand, if previously we had c F (x) = c F (y) = i, and WLOG x ∈ B i , then the definition of c(v) gives that we now have that c(x) ∈ {i − 1, i + 1} (modulo 2ℓ − 1). Thus if c is not a good C 2ℓ+1 -coloring of S, then there is an edge xy ∈ M such that x ∈ B i and y's color also changes in the coloring c; but by the distance between edges in M, this can only happen if x and y are at F -distance < 2ℓ − 1. Note also that c F (x) = c F (y) implies that dist F (x, y) is even. Thus in this case, F ∪ {xy} contains an odd cycle of length ≤ 2ℓ − 1, and so G has odd girth < 2ℓ + 1, as desired.

Avoiding homomorphisms to long odd cycles
For large ℓ, one can prove the non-existence of homomorphisms to C 2ℓ+1 using the following simple observation: Observation 4. If G has a homomorphism to C 2ℓ+1 , then G has an induced bipartite subgraph with at least 2ℓ 2ℓ+1 |V (G)| vertices.
Proof. Delete the smallest color class.
Proof of Theorem 2. The probability that G n, c n has an induced bipartite subgraph on βn vertices is at most n βn 2 βn 1 − c n The expression inside the parentheses is unimodal in β for fixed c, and, for c > 2.774, is less than 1 for β > .999971. In particular, for c > 2.774, G n, c n has no homomorphism to C 2ℓ+1 for 2ℓ + 1 ≥ 1, 427, 583. P2 There are no edges between V i and V i+2 ∪ V i−2 . Here addition and subtraction in an index are taken to be modulo 5.
P4 Every v ∈ V 2 has a neighbor in V 3 .
Hatami [6], Lemma 2.1 shows that we can assume P1,P2,P3. Given P1,P2,P3, if v ∈ V 2 has no neighbors in V 3 then we can move v from from V 2 to V 0 and still have a homomorphism. Furthermore, this move does not upset P1,P2,P3.
We let |V i | = n i for i = 0, 1, . . . , 4. For a fixed partition we then have Equations (5) and (6) are self evident, but we need to justify (7). Consider the bipartite subgraph Γ of G n,p induced by V 2 ∪ V 3 . P3 tells us that each v ∈ V 3 has a neighbor in V 2 . Denote this event by A. Suppose now that we choose a random mapping φ from V 3 to V 2 . We then create a bipartite graph Γ ′ with edge set E 1 ∪E 2 . Here E 1 = {xy : x ∈ V 3 , y = φ(x)} and E 2 is obtained by independently including each of the n 2 n 3 possible edges between V 2 and V 3 with probability p. We now claim that we can couple Γ, Γ ′ so that Γ ⊆ Γ ′ .
Event A can be construed as follows: A vertex in v ∈ V 3 chooses B v neighbors in V 2 where B v is distributed as a binomial Bin(n 2 , p), conditioned to be at least one. The neighbors of v in V 2 will then be a random B v subset of V 2 . We only have to prove then that if v chooses B ′ v random neighbors in Γ ′ then B ′ v stochastically dominates B v . But B ′ v is one plus Bin(n 2 − 1, p) and domination is easy to confirm. We have n 2 − 1 instead of n 2 , since we do not wish to count the edge v to φ(v) twice.
We now write n i = α i n for i = 0, . . . , 4. We are particularly interested in the case where c = 4. Now (4) implies that G n, 4 n has no induced bipartite subgraph of size βn for β > 0.94. Thus we may assume that α i ≥ 0.06 for i = 0, . . . , 4. In which case we can write The number of choices for V 0 , . . . , V 4 with these sizes is Putting In the next section, we describe a numerical procedure for verifying that the maximum in (8) is less than 1. This will complete the proof of Theorem 3.
6 Bounding the function.
Our aim now is to bound the partial derivatives of b(4.0, α 0 , α 1 , α 2 , α 3 ), to translate numerical computations of the function on a grid to a rigorous upper bound.
In the calculations below we will make use of the following bounds: They assume that 0.06 ≤ α i ≤ 0.6 for i ≥ 0.