A note on coloring vertex-transitive graphs

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.


Introduction
Many results and conjectures in the graph coloring literature have the form: if the chromatic number χ of a graph is close to its maximum degree ∆, then the graph contains a big clique, i.e., ω is large ( [3,2,22,21,4,16]). Generically, we call conjectures of this sort big clique conjectures. In [20], it was shown that many big clique conjectures hold under the added hypothesis that every vertex is in a medium sized clique. Partial results on big clique conjectures often guarantee a medium sized clique, but not a big clique. But in a vertextransitive graph, the existence of one medium sized clique implies that every vertex is in a medium sized clique. By applying the idea in [20], we now get a big clique. So, in essence, partial results on big clique conjectures are self-strengthening in the class of vertex-transitive graphs.
In this short note, we give some examples of this phenomenon. There is not much new graph theory here, just combinations of known results that yield facts we did not know. The following conjecture is the best we could hope for. A good deal of evidence supports it, as we will detail below.
Our Main Conjecture would be best possible, as shown by Catlin's counterexamples to the Hajós conjecture [5]. Catlin computed the chromatic number of line graphs of odd cycles where each edge has been duplicated k times; in particular, he showed that χ(G t,k ) = 2k+ k for t ≥ 2, where G t,k := L(kC 2t+1 ). Since ∆(G t,k ) = 3k − 1 and ω(G t,k ) = 2k, we have for all k ≥ 1. The Main Conjecture does not hold for graphs in general. To see this, let H t be K t joined to a 5-cycle. For any c ≥ 2, we can make t large enough so that χ(H t ) > max {ω(H t ), ∆(H t ) − c} and ∆(H t ) is as large as we like. So, no bound of this form can hold in general, not even for claw-free graphs (since H t is claw-free). In [17], it was shown that a bound of this form does hold for line graphs of multigraphs. In particular, they satisfy χ ≤ max ω, 7∆+10 8 . The bound in the Main Conjecture would be best possible for line graphs of multigraphs as well.
Our main result is the following weakening of the Borodin-Kostochka conjecture for vertex-transitive graphs, which we prove in Section 5. This theorem likely holds for all ∆ ≥ 9 and proving this may be a good deal easier than proving the full Borodin-Kostochka conjecture (note that the Main Conjecture implies the Main Theorem for all ∆ ≥ 9).
As further evidence for the Main Conjecture, we show that the analogous upper bound holds for the fractional chromatic number. Also, we show that the Main Conjecture is true if all vertex-transitive graphs satisfy both Reed's ω, ∆, and χ conjecture and the strong 2∆-colorability conjecture (see [1]; really we can get by with 5 2 ∆-colorability). Finally, we show the following.

Clustering of maximum cliques
Before coloring anything, we need a better understanding of the structure of maximum cliques in a graph.

The clique graph
Definition 1. Let G be a graph. For a collection of cliques Q in G, let X Q be the intersection graph of Q; that is, the vertex set of X Q is Q and there is an edge between Q 1 , Q 2 ∈ Q iff Q 1 = Q 2 and Q 1 and Q 2 intersect.
When Q is a collection of maximum cliques, we get a lot of information about X Q . Kostochka [15] used the following lemma of Hajnal [10] to show that the components of X Q are complete in a graph with ω > 2 3 (∆ + 1). Lemma 2.1 (Hajnal [10]). If G is a graph and Q is a collection of maximum cliques in G, Hajnal's lemma follows by an easy induction. The proof of Kostochka's lemma in [15] is in Russian; for a reproduction of his original proof in English, see [18]. Below we give a shorter proof from [19]. [15]). If Q is a collection of maximum cliques in a graph G with ω(G) > 2 3 (∆(G) + 1) such that X Q is connected, then ∩Q = ∅. Proof. Suppose not and choose a counterexample Q := {Q 1 , . . . , Q r } minimizing r. Plainly, r ≥ 3. Let A be a noncutvertex in X Q and B a neighbor of A. Put Z := Q − {A}. Then X Z is connected and hence by minimality of r, ∩Z = ∅. In particular, |∪Z| ≤ ∆(G) + 1.
As shown by Christofides, Edwards and King [6], components of X Q have nice structure in the ω = 2 3 (∆ + 1) case as well. We'll need this stronger result to get our bounds on coloring vertex-transitive graphs to be tight.

In vertex-transitive graphs
Let G be a vertex-transitive graph and let Q be the collection of all maximum cliques in G.
It is not hard to see that X Q is vertex-transitive as well; in fact, we have the following.
Observation 1. Let G be a vertex-transitive graph and let Q be the collection of all maximum cliques in G.
A basic consequence of Observation 1 is that if G is vertex-transitive and G C has a dominating vertex (or universal vertex), then every vertex of G C is dominating; so G C is complete. Let G be a vertex-transitive graph with ω > 2 3 (∆ + 1). Suppose that X Q has one or more edges. By Kostochka's lemma, ∩Q C is nonempty, where Q C is the set of maximum cliques in some component G C . Choose a vertex v ∈ ∩Q C , and note that v is adjacent to each vertex in G C . Since G is vertex-transitive, each vertex of G C is a dominating vertex in G C ; so, in fact, G C is a clique, and C is edgeless. Using Lemma 2.3, we get a bit more.
Lemma 2.4. Let G be a connected vertex-transitive graph and let Q be the collection of all maximum cliques in G. If ω(G) ≥ 2 3 (∆(G) + 1), then either • X Q is edgeless; or • X Q is a cycle and G is the graph obtained from X Q by blowing up each vertex to a K 1 2 ω(G) .
Proof. If ω(G) > 2 3 (∆(G) + 1), then X Q is edgeless as shown above. Hence we may assume ω(G) = 2 3 (∆(G) + 1). Let Z be a component of X Q and put Z := V (Z). By Lemma 2.3, ∆(X Z ) ≤ 2 and if B, C ∈ Z are different neighbors of A ∈ Z, then B ∩ C = ∅ and |A ∩ B| = |A ∩ C| = 1 2 ω(G). By Observation 1, X Z must be a cycle. But then every vertex in G Z has 1 2 ω(G) + 1 2 ω(G) + 1 2 ω(G) − 1 = ∆(G) neighbors in G Z and thus G = G Z . Hence X Q = X Z is a cycle and G is the graph obtained from X Q by blowing up each vertex to a K 1 2 ω(G) .

The fractional version
The problem of determining chromatic number can be phrased as an integer program: we aim to minimize the total number of colors used, subject to the constraints that (i) each vertex gets colored and (ii) the vertices receiving each color form an independent set. To reach a linear program from this integer program, we relax the constraint that each vertex is colored with a single color, and instead allow a vertex to be colored with a combination of colors, e.g., 1/2 red, 1/3 green, and 1/6 blue. However, we still require that the total weight of any color on any clique is at most 1. The minimum value of this linear program is the fractional chromatic number, denoted χ f (see [23] for a formal definition and many results on fractional coloring). It is an easy exercise to show that every vertex-transitive graph G satisfies χ f (G) = |G| α(G) , where |G| denotes |V (G)| and α(G) denotes the maximum size of an independent set. We also need Haxell's condition [11] for the existence of an independent transversal.  Proof. We may assume that G is connected. Since G is vertex-transitive, every vertex of G is in an ω(G)-clique. First, suppose ω(G) > 2 3 (∆(G) + 1). Then Lemma 2.4 shows that the vertex set of G can be partitioned into cliques V 1 , . . . , V r with |V i | ≥ 2 3 (∆(G) + 1) for each i ∈ [r]. Let H be the graph formed from G by making each V i independent. Then ∆(H) ≤ ∆(G) + 1 − 2 3 (∆(G) + 1) ; now by Lemma 3.1, G has an independent set with a vertex in each V i . Since G is vertex-transitive, all V i have the same size; so, in fact, So instead suppose ω(G) = 2 3 (∆(G) + 1). Now Lemma 2.4 shows that G is obtained from a cycle C by blowing up each vertex of C to a copy of K 1 2 ω(G) . Hence α(G) = |C| as desired.
Reed's ω, ∆, and χ conjecture states that every graph satisfies In [16], Molloy and Reed proved this upper bound without the round-up for the fractional chromatic number χ f . Since χ f (G) = |G| α(G) for vertex-transitive graphs, an earlier result of Fajtlowicz [8] suffices for our purposes. as desired.
Restating Theorem 3.4 in terms of fractional coloring, we have the following.

Reed's conjecture plus strong coloring
For a positive integer r, a graph G with |G| = rk is called strongly r-colorable if for every partition of V (G) into parts of size r there is a proper coloring of G that uses all r colors on each part. If |G| is not a multiple of r, then G is strongly r-colorable iff the graph formed by adding r |G| r − |G| isolated vertices to G is strongly r-colorable. The strong chromatic number sχ(G) is the smallest r for which G is strongly r-colorable. Not surprisingly, if G is strongly r-colorable, then G is also strongly (r + 1)-colorable, although the proof of this fact is non-trivial [9]. In [12], Haxell proved that the strong chromatic number of any graph is at most 3∆−1. In [13], she proved further that for every c > 11/4 there exists ∆ c such that if G has maximum degree ∆ at least ∆ c , then G has strong chromatic number at most c∆. The strong 2∆colorability conjecture [1] says that the strong chromatic number of any graph is at most 2∆. If true, this conjecture would be sharp. We need the following intermediate conjecture. We also need Reed's conjecture [21] restricted to vertex-transitive graphs. Proof. We may assume that G is connected. Put ∆ := ∆(G), ω := ω(G) and χ := χ(G).
Suppose ω < 2 3 (∆ + 1). So, we have ω ≤ 2∆+1 3 and moreover, when ∆ ≡ 3 (mod 6), we have ω ≤ 2 3 ∆. Plugging the first inequality into Conjecture 4.2 gives χ ≤ 5∆+4 6 = 5∆+3 6 when ∆ ≡ 3 (mod 6); by using the improved upper bound on ω in the remaining case, we again prove the desired upper bound on χ. Now suppose ω ≥ 2 3 (∆ + 1) and let Q be the set of maximum cliques in G. Applying Lemma 2.4, either X Q is edgeless or G is obtained from an odd cycle by blowing up each vertex to a K ω 2 . In the latter case, G is one of Catlin's examples from [5] and the bound holds as mentioned in the introduction. Hence we may assume that X Q is edgeless; that is, V (G) can be partitioned into ω(G)-cliques.
Suppose χ > ω. Now we show that Conjecture 4.1 implies the Main Conjecture. Form G ′ from G by adding vertices to the maximum cliques of G until they all have 5∆+3 6 vertices; each new vertex has no edges outside its clique, and ∆ always denotes the maximum degree in G, not in G ′ . Now form G ′′ from G ′ by removing all edges within each maximum clique. Each vertex now has at most ∆+1−ω ≤ 1 3 (∆+1) neighbors in G ′ outside of its clique, hence the maximum degree of G ′′ is at most 1 3 (∆ + 1). Since 5∆+3 6 ≥ 5 2 1 3 (∆ + 1) , Conjecture 4.1 implies that G ′′ is strongly 5∆+3 6 -colorable. By taking the V i 's of G ′′ to be the vertex sets of the maximum cliques in G ′ , we see that G ′ is 5∆+3 6 -colorable, and hence so is G.

Borodin-Kostochka for vertex-transitive graphs
In [7], we proved the following. In [20], the second author proved the following.