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Coloring triangle‐free graphs with local list sizes

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Molloy's result actually guarantees a proper coloring of the graph in the more general situation that every vertex is supplied permissible color lists of size ⌈(1 + )Δ∕ log Δ⌉. It is natural to ask what happens if fewer colors are supplied to vertices that are not of maximum degree; indeed one might expect the low degree vertices to be easier to color in a quantifiable way.
The general idea of having "local" list sizes is far from new; it can be traced at least back to degree-choosability as introduced in one of the originating papers for list coloring [8]. Recently Bonamy, Kelly, Nelson, and Postle [4] initiated a modern and rather general treatment of this idea, including with respect to triangle-free graphs. (A conjecture of King [12] and related work are in the same vein.) We show the following result. Theorem 1. Fix > 0, let Δ be sufficiently large, and let = (192 log Δ) 2∕ . Let G be a triangle-free graph of maximum degree Δ and L ∶ V(G) → 2 Z + be a list assignment of G such that for all v ∈ V(G), , log } .

Then there exists a proper coloring c ∶ V(G) → Z + of G such that c(v) ∈ L(v) for all v ∈ V(G).
This of course implies Molloy's theorem, and can be considered a local strengthening. When the graph G in Theorem 1 is of minimum degree , the list size condition is local in the sense that the lower bound on |L(v)| reduces to a function of deg (v) and no other parameter of G. Theorem 1 (or rather the stronger Theorem 8) improves upon [4,Thm. 1.12], by having an asymptotic leading constant of 1 rather than 4 log 2, at the expense of requiring a larger minimum list size. Our proof relies heavily on the work of Bernshteyn [3], who gave a further simplified proof for a stronger version of Molloy's theorem. For Theorem 1, it has sufficed to prove a local version of the so-called "finishing blow" (see Lemma 6) and to notice that there is more than enough slack in Bernshteyn's (and indeed Molloy's) argument to satisfy the new blow's hypothesis. We also provide a local version of Molloy's theorem for a relaxed, fractional form of coloring. Writing (G) for the set of independent sets of G, and for the standard Lesbegue measure on R, a fractional coloring of a graph G is an assignment w(I) for I ∈ (G) of pairwise disjoint measurable subsets of R to independent sets such that ∑ I∈(G),I∋v (w(I)) ≥ 1 for all v ∈ V(G). Such a coloring naturally induces an assignment of measurable subsets to the vertices of G, namely w(v) = ⋃ I∈(G),I∋v w(I) for each v ∈ V(G), such that w(u) and w(v) are disjoint whenever uv ∈ E(G). The total weight of the fractional coloring isŵ(G) = ∑ I∈(G) (w(I)).

Theorem 2.
For all > 0 there exists > 0 such that every triangle-free graph G admits a fractional coloring w such that for every v ∈ V(G) Again, when G is of minimum degree our condition on w(v) reduces to a function of deg(v) alone, yielding a local condition. Clearly Theorem 2 is not implied by Molloy's theorem or is the converse true, but both results imply that the fractional chromatic number of a triangle-free graph of maximum degree Δ is at most (1 + o(1))Δ∕ log Δ. We believe that the main interest in Theorem 2 will be in its derivation. We give a short and completely self-contained proof by analyzing a probability distribution on independent sets known as the hard-core model in triangle-free graphs (Lemma 4), and demonstrating that to obtain the desired result it suffices to feed this distribution as input to a greedy fractional coloring algorithm (Lemma 3). Since it makes no use of the Lovász Local Lemma, the proof is unlike any other derivation of a Johansson-type coloring result (regardless of local list sizes). This may be of independent interest. The asymptotic leading constant of 1 in the conditions of both Theorems 1 and 2 cannot be improved below 1/2 due to random regular graphs [10]. In fact, as a corollary of either result we match asymptotically the upper bound of Shearer [15] for off-diagonal Ramsey numbers. So any improvement below 1, or even to 1 precisely (i.e., removal of the term), would be a significant advance. To give more detail, Shearer proved 1 that as Δ → ∞ any triangle-free graph on n vertices of maximum degree Δ contains an independent set of size at least It is easy to show that any graph contains an independent set of size at least n∕ if it permits any of a fractional coloring with total weight , a proper coloring with colors, or an L-coloring whenever |L(v)| ≥ for all vertices v, and hence the leading constant in the bound of Molloy, and in Theorems 1 and 2 cannot be improved without improving this "Shearer bound" on the independence number of triangle-free graphs. Our analysis of the hard-core model on triangle-free graphs has its roots in [7], where the first author, Jenssen, Perkins, and Roberts showed that for G as above, (1) is a lower bound on the expected size of an independent set from the hard-core model (when a parameter known as the fugacity is not too small). Most intriguingly, they proved that their result is asymptotically tight by appealing to the random regular graph, whereas Theorem 2 is not known to be tight: there is a factor two gap between the fractional chromatic number of the random regular graph and the bound one gets via Theorem 2 or Molloy's result. We define the hard-core model and the fugacity parameter in Section 3. Allow us to make some further remarks related to the maxima that occur in the list and weight conditions of Theorems 1 and 2, which at first sight seem artificial and unnecessary. In Theorem 2 the parameter is a function of alone but in Theorem 1 we require to grow with Δ. So for large enough Δ the value of in Theorem 2 is strictly smaller 2 than the value in Theorem 1, and since the list chromatic number can be much larger than the fractional chromatic number (even for bipartite graphs) neither of Theorems 1 and 2 implies the other. In Section 7, we show that these are truly distinct results in that, unlike in Theorem 2, some nontrivial (albeit very slight) dependence between minimum list size and maximum degree is necessary in Theorem 1. Last observe that, if we were able to improve either result by lowering to a quantity independent of , then it would constitute a significant improvement over Shearer's bound.
We are hopeful that some of the techniques we used in this paper might also be applicable to other natural coloring problems in triangle-free graphs, such as bounding the (list) chromatic number in terms of the number of vertices, cf. [5, Conjs. 4.3 and 6.1], but leave this for further investigation.

Structure of the paper
In Section 2, we prove a greedy fractional coloring lemma (Lemma 3). We give a local analysis of the hard-core model in triangle-free graphs in Section 3, culminating in Lemma 4. As a demonstration of its further applicability, we also use Lemma 4 to give a good bound on semi-bipartite induced density in triangle-free graphs (Theorem 5), a concept related to a recent conjecture of Esperet, Thomassé and the third author [9]. We prove Theorem 2 in Section 4. In Section 5, we review the definition of correspondence coloring and prove for it a local version of the "finishing blow" (Lemma 6). In Section 6 we sketch how Bernshteyn's argument can then be adapted to prove Theorem 1. In Section 7, we present a simple construction (Proposition 11) to show that even some bipartite graphs cannot satisfy the conclusions of Theorem 1 without a suitable lower bound on .

Notation and preliminaries
For a graph G and vertex v ∈ V(G), we write N G (v) for the set of neighbors of v in a graph G, and We have already indicated above that (G) denotes the set of independent sets of G. Note that ∅ ∈ (G) for all G.
The function W is the inverse of z  → ze z , also known as the Lambert W-function, which satisfies We will have use for the following probabilistic tool, see [2].

The General Lovász Local Lemma. Consider a set
then the probability that no event in  occurs is at least

A FRACTIONAL COLORING ALGORITHM
The following result for local fractional coloring is slightly stronger than what we require in the proof of Theorem 2, but the proof is no different from that needed for the weaker statement.

Then there exists a fractional coloring of G such that every v ∈ V(G) is colored with a subset of the interval
Proof. We present a refinement of an algorithm given in the book of Molloy and Reed [14], and show that under the assumptions of the lemma, it returns the desired fractional coloring. The idea of the algorithm is to greedily add weight to independent sets according to the probability distribution induced on all not yet fully colored vertices. For brevity, we write We build a fractional coloring w in several iterations, and we writeŵ(I) for (w(I)) so thatŵ(I) is a nonnegative integer representing the measure w assigns to I. Through the iterations, w is a partial fractional coloring in the sense of not yet having satisfied the condition that ∑ I∈(G),I∋vŵ (I) ≥ 1 for all v ∈ V(G). We extend our notational conventions for w toŵ, so thatŵ(G) = ∑ I∈(G)ŵ (I) is the total measure used by the current partial coloring, andŵ(v) = ∑ I∈(G),I∋vŵ (I) for any v ∈ V(G) is the total measure given to a vertex v by the current partial coloring.

Algorithm 1
The greedy fractional coloring algorithm We next show that Algorithm 1 certifies the desired fractional coloring. For the analysis, it is convenient to index the iterations: for i = 0, 1, … , let H i ,ŵ i (I),ŵ i (v),ŵ i (G), i denote the corresponding H,ŵ(I),ŵ(v),ŵ(G), in the ith iteration prior to updating the sequence. Note then that Let us first describe the precise fractional coloring (rather than its sequence of measures) that is constructed during Algorithm 1. During the update fromŵ i toŵ i+1 , in actuality we do the following. Divide the interval [ŵ i (G),ŵ i (G)+ i ) into a sequence (B I ) I∈(G) of consecutive right half-open intervals such that B I has length P(I H i = I) i . We then let w i+1 (I) = w i (I) ∪ B I for each I ∈ (G). Note that (w i (I)) =ŵ i (I) for all I ∈ (G) and i. Moreover, by induction, . So we only need to show that Algorithm 1 terminates. To do so, it suffices to we are done. We may therefore assume that there is some For any k ∈ {0, … , i}, we know that By summing this last inequality over all such k, we obtain and (v) is defined to be the left-hand side of this chain of inequalities, so we have equality throughout. We also note that because for any u appearing in the sum on the right-hand side of the first line we have 0 ≤ŵ i+1 (u) ≤ 1. Indeed, the choice of i in the algorithm ensures that the weightsŵ i+1 (u) never exceed 1. We then have from (2) that and hence thatŵ i+1 (v) = 1. This means |V(H i+1 )| < |V(H i )|, as required for a proof of termination. ▪

A LOCAL ANALYSIS OF THE HARD-CORE MODEL
Given a graph G, and a parameter > 0, the hard-core model on G at fugacity is a probability distribution on the independent sets (G) (including the empty set) of G, where each I ∈ (G) occurs with probability proportional to |I| . Writing I for the random independent set, we have where the normalizing term in the denominator is the partition function (or independence polynomial) Given a choice of I ∈ (G), we say that a vertex u ∈ V(G) is uncovered if N(u) ∩ I = ∅, and that u is occupied if u ∈ I. Note that u can be occupied only if it is uncovered.
For the rest of this section we assume that G is triangle-free. We note the following useful facts (which appear verbatim in [6,7]).
We apply these facts to give a lower bound on a linear combination of the probability that v is occupied and the expected number of occupied neighbors of v. This is a slight modification of the arguments of [6,7], but here we focus on individual vertices, rather than averaging over a uniformly random choice of vertex.

Lemma 4. Let G be a triangle-free graph and let ( v ) v∈V(G) and ( v ) v∈V(G) be sequences of positive real numbers. Write I for a random independent set drawn from the hard-core model on G at fugacity
Proof. Fix a vertex v ∈ V(G) and let Z be the number of uncovered neighbors of v given the random independent set I. By Fact 1, conditioning on the number of uncovered neighbors of v, and by Fact 2, we have where for the final inequality we used Jensen's inequality. Similarly, each of the Z uncovered neighbors of v is occupied with probability ∕(1 + ) independently of the others (since G is triangle-free), and a covered neighbor of v is occupied with probability zero. Hence Then for any vertex v ∈ V(G) and positive reals v and v we have and since EZ is some (nonnegative) real number we also have When v , > 0 the function g is strictly convex (because its second derivative is positive), and hence has a unique stationary point at z = z * , say, which gives its minimum. We compute that showing that for every vertex We next give a result related to a recent conjecture of Esperet, Thomassé and the third author [9, Conj. 1.5]. A semi-bipartite induced subgraph of a graph G is a subgraph H of G consisting of all edges between two disjoint subsets A, B ⊂ V(G) such that A is independent. This definition means that average degree of such a semi-bipartite induced subgraph H is 2 |A|+|B| e G (A, B), where e G (A, B) represents the number of edges of G with one endpoint in A and one endpoint in B. Our local analysis of the hard-core model in triangle-free graphs yields a semi-bipartite induced subgraph of high average degree, measured by a property of G that incorporates local degree information: the geometric mean of the degree sequence. We improve upon [9,Thm. 3.5] by replacing minimum degree with the geometric mean of the degrees, and increasing the leading constant.

Theorem 5. A triangle-free graph G on n vertices contains a semi-bipartite induced subgraph of average degree at least
In the statement of the theorem and in the proof below, the o(1) term tends to zero as the geometric mean of the degree sequence of G tends to infinity.
Proof of Theorem 5. We find a semi-bipartite induced subgraph of G where one of the parts is a random independent set I from the hard-core model, and the other is V(G) ⧵ I. The number of edges between the parts is therefore e G (I, V(G) ⧵ I) = ∑ v∈I deg(v), which is a random variable we denote X. We write EX in two different ways: The first version follows from linearity of expectation, and for the second we note that E|N(v) ∩ I| = ∑ u∈N(v) P(u ∈ I) and hence P(u ∈ I) appears deg(u) times in the sum as required. For brevity, we write ∑ v for a sum over v ∈ V(G) in the rest of the proof. Then for any , > 0 we have hence by Lemma 4, Choosing, for example, ∕ = = n∕ ∑ v log deg(v), we observe that To complete the proof, note that the bound on EX means that there is at least one independent set I with at least (1 + o(1)) ∑ v log deg(v) edges from I to its complement. This immediately means that the average degree of the semi-bipartite subgraph with parts I and V(G) ⧵ I is at least We remark that the methods of [7] deal with the quantities P(v ∈ I) and E|N(v) ∩ I| in a slightly more sophisticated manner that avoids the seemingly arbitrary parameter ∕ in the above proof. Since we have Lemma 4 for other purposes in this paper, it is expedient to use it here.

LOCAL FRACTIONAL COLORING
Proof of Theorem 2. The method is to combine Lemmas 3 and 4 by carefully choosing For then the hypothesis of Lemma 3 (with 0 (v) = v and 1 (v) = v for all v ∈ V(G)) follows from the conclusion of Lemma 4. Given the assumptions on G, we can apply Lemma 4 to any induced subgraph H of G since such H are also triangle-free and the local parameters v and v are invariant under taking induced subgraphs.

Note that (3) is equivalent to
For any fixed this is an increasing function of deg(v). We take = ∕2, and we are done by Lemma 3 if we can show that there exists > 0 such that for all deg(v) ≥ we have Let us first assume that deg(v) is at least some large enough multiple of 1∕ so that where we used the fact that W(x) = log x − log log x + o(1) as x → ∞. Then by (4), it suffices to have that (1 + ∕2)).
This last inequality holds for deg(v) large enough (as a function of ) provided This is easily checked to hold true for small enough , namely ≤ 4. ▪

A LIST COLORING LEMMA
Just as in [3], we will establish Theorem 1 for a generalized form of list coloring called correspondence coloring (or DP-coloring). We here state the definition given in [3]. Given a graph G, a cover of G is a pair ℋ = (L, H), consisting of a graph H and a function L ∶ V(G) → 2 V(H) , satisfying the following requirements: An ℋ-coloring of G is an independent set in H of size |V(G)|. A reader who prefers not to concern herself with this generalized notion may merely read L as an ordinary list assignment and V(H) as the disjoint union of all lists. For usual list coloring, there is an edge in H between equal colors of two lists if and only if there is an edge between their corresponding vertices in G.
To state and prove our local version of the finishing blow, we will need some further notation. For clarity, we separately state the corollary this lemma has for conventional list coloring.  Note that, if I ′ is an ℋ I -coloring of G I , then I ∪ I ′ is an ℋ-coloring of G.
With this notation, and in view of Lemma 6, it suffices to establish the following analogue of Lemma 3.5 in [3]. In exactly the same way that Lemma 3.5 in [3] follows from Lemma 3.6 in [3], Lemma 9 follows from the following result. We refer the reader to [3] for further details.
Proof sketch. Since the proof is nearly the same as the proof of Lemma 3.6 in [3], we only highlight the essential differences. The two proofs are completely identical until the application of Jensen's Inequality ("by the convexity…"), where we instead get where the final inequality holds for Δ (and hence ) large enough in terms of , because by convexity 1 − 1∕(1 + ) > ∕2 for 0 < < 1. The application of a Chernoff bound for negatively correlated random variables applies in the same way as in Bernshteyn's proof to yield that which is at most Δ −3 ∕8 for Δ ≥ 2.
For the second part of the proof, we instead for all c ∈ L(u) define and it will suffice to show p c ≤ Δ −4 . The argument is the same to show that for Δ large enough in terms of , and a similar second application of a Chernoff bound then yields as required. ▪

A NECESSARY MINIMUM DEGREE CONDITION FOR BIPARTITE GRAPHS
In Theorem 1 the condition is only truly local when the graph is of minimum degree = (192 log Δ) 2∕ , which grows with the maximum degree Δ. The result is made strictly stronger by reducing . In this section we show that even for bipartite graphs the conclusion of Theorem 1 requires some (1) bound on as Δ → ∞. We state and prove the result specifically with deg(u)∕ log deg(u) as the target local list size per vertex u. The reader can check that any sublinear and superlogarithmic function will do, but with a different tower of exponentials. Proposition 11. For any , there is a bipartite graph of minimum degree and maximum degree exp −1 ( ) (so a tower of exponentials of height − 1) that is not L-colorable for some list assignment L ∶ V(G) → 2 Z + satisfying |L(u)| ≥ deg(u) log deg (u) for all u ∈ V(G).
Proof. The construction is a recursion, iterated − 1 times.
For the basis of the recursion, let G 0 be the star K 1, of degree . We write A 0 as the set containing the center v 0 of the star and B 0 as the set of all noncentral vertices. Note that, with the assignment L 0 that assigns the list {1 0 , … , 0 } to the center and lists {i 0 }, i ∈ [ ], to the noncentral vertices, G 0 is not L 0 -colorable.
We recursively establish the following properties for G i , A i , B i , L i , where 0 ≤ i ≤ − 1: (1) G i is bipartite with partite sets A i and B i ; (2) A i has all vertices of degree at least and at most exp i ( ), with some vertex v i attaining the maximum exp i ( ); (3) B i has exp i ( ) vertices of degree i + 1; (4) |L i (a)| ≥ deg(a)∕ log deg(a) for all a ∈ A i and |L i (b)| ≥ deg(b) for all b ∈ B i ; and (5) G i is not L i -colorable.
These properties are clearly satisfied for i = 0.
From step i to step i + 1, we form G i+1 by taking exp(exp i ( ))∕ exp i ( ) copies of G i and adding a vertex v i+1 universal to all of the B i -vertices. Let A i+1 be v i+1 together with all A i -vertices, and B i+1 be all of the B i -vertices. Label each copy of G i with j from 1 to exp(exp i ( ))∕ exp i ( ). We set L i+1 (v i+1 ) = {1 i+1 , … , exp(exp i ( ))∕ exp i ( ) i+1 } and add color j i+1 to L i (b) to form L i+1 (b) for every B i -vertex b in the jth copy of G i . It is routine to check then that G i+1 , A i+1 , B i+1 , L i+1 satisfy the promised properties.
The proposition follows by taking G −1 . ▪ As a final remark on minimum degree or minimum list size conditions, we note that our proof of Theorem 1 can be adapted to reduce = (192 log Δ) 2∕ as a function of Δ by increasing the leading constant "1" in the list size condition. Indeed, this removes the dependence on and brings the result much closer to the triangle-free case of the significantly more general local coloring result of Bonamy, Kelly, Nelson, and Postle [4], which has a minimum degree condition of (log Δ) 2 . Here, as we focus on triangle-free graphs we prefer to aim for the best possible constant at the expense of the cutoff value .