Separating Polynomial $\chi$-Boundedness from $\chi$-Boundedness

Abstract

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}$ with $f(1)=1$ and $f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right)$, we construct a hereditary class of graphs ${\mathcal {G}}$ such that the maximum chromatic number of a graph in ${\mathcal {G}}$ with clique number n is equal to f(n) for every $n\in \mathbb {N}$. In particular, we prove that there exist hereditary classes of graphs that are $\chi$-bounded but not polynomially $\chi$-bounded.

1 Introduction

Given a class of graphs ${\mathcal {C}}$ its $\chi$-bounding function is the function $\chi _{\mathcal {C}}:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}$ defined as

$$\begin{aligned} \chi _{\mathcal {C}}(n)=\sup \{\chi (G):G\in {\mathcal {C}}\text { and } \omega (G)=n\} \text {,} \end{aligned}$$

where $\chi (G)$ and $\omega (G)$ denote, respectively, the chromatic number and the clique number of G. A class of graphs ${\mathcal {C}}$ is $\chi$-bounded if there is a function $f:\mathbb {N}\rightarrow \mathbb {N}$ such that $\chi (G)\leqslant f(\omega (G))$ for every graph $G\in {\mathcal {C}}$, or equivalently if $\chi _{\mathcal {C}}(n)$ is finite for every $n\in \mathbb {N}$. A class ${\mathcal {C}}$ is polynomially $\chi$-bounded if such a function f can be chosen to be a polynomial. A class ${\mathcal {C}}$ is hereditary if it is closed under taking induced subgraphs.

A well-known and fundamental open problem, due to Esperet [6], has been to decide whether every hereditary class of graphs which is $\chi$-bounded is polynomially $\chi$-bounded. We provide a negative answer to this question. More generally, we prove that $\chi$-bounding functions may be arbitrary, so long as they are bounded from below by a certain cubic function.

Theorem 1

Let $f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}$ be such that $f(1)=1$ and $f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right)$ for every $n\geqslant 2$. Then there exists a hereditary class of graphs ${\mathcal {G}}$ such that $\chi _{\mathcal {G}}(n)=f(n)$ for every $n\in \mathbb {N}$.

On the other hand, $\chi$-bounding functions are not entirely arbitrary. For instance, Scott and Seymour [11] proved that every hereditary class of graphs ${\mathcal {C}}$ with $\chi _{\mathcal {C}}(2)=2$ satisfies $\chi _{\mathcal {C}}(n)\leqslant 2^{2^{n+1}}$.

The proof of Theorem 1 is heavily based on the idea used by Carbonero, Hompe, Moore, and Spirkl [2] in their very recent solution to another well-known problem attributed to Esperet [12]. They proved that for every $k\in \mathbb {N}$, there is a graph G with $\omega (G)=3$ and $\chi (G)\geqslant k$ such that every triangle-free induced subgraph of G has chromatic number at most 4. Their proof, in turn, relies on an idea by Kierstead and Trotter [8], who proved in 1992 that the class of oriented graphs excluding a directed path on four vertices as an induced subgraph is not $\chi$-bounded. We further generalise the aforesaid result of Carbonero, Hompe, Moore, and Spirkl [2] to higher clique numbers. Specifically, we prove the following general bound, which we use to derive Theorem 1.

Theorem 2

For every pair of integers n and k with $k\geqslant n\geqslant 2$, there exists a graph G with clique number n and chromatic number k such that every induced subgraph of G with clique number $m<n$ has chromatic number at most $\left( {\begin{array}{c}3m+1\\ 3\end{array}}\right)$.

In case that n is a prime number, we prove a better bound, which matches the bound of 4 from [2] when $n=3$.

Theorem 3

For every pair of integers p and k with p a prime and $k\geqslant p$, there exists a graph G with clique number p and chromatic number k such that every induced subgraph of G with clique number $m<p$ has chromatic number at most $\left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$.

In the first version of this paper [1], we proved a weaker version of Theorem 3 with $\left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$ replaced by $m^{m^2}$. Despite the worse bound obtained, that alternative proof may still be of interest. The mere qualitative statement that for every prime p, there are graphs with clique number p and arbitrarily large chromatic number whose induced subgraphs with clique number less than p have bounded chromatic number suffices to imply the negative answer to Esperet’s question.

After [1] appeared, Girão et al. [7] proved another generalisation of the aforesaid qualitative version of Theorem 3. Namely, they proved that for every graph F with at least one edge, there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F-free induced subgraph has chromatic number at most some constant $c_F$ depending only on F. They also showed the analogous statement where clique number is replaced by odd girth.

See [12] and [9] for recent surveys on $\chi$-boundedness and polynomial $\chi$-boundedness.

2 Proof

First, we show that Theorem 2 implies Theorem 1.

Proof of Theorem 1 Assuming Theorem 2

Fix a function $f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}$ such that $f(1)=1$ and $f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right)$ for every $n\geqslant 2$. By Theorem 2, for every pair of integers n and k with $k\geqslant n\geqslant 2$, there exists a graph $H_{n,k}$ with clique number n and chromatic number k such that every induced subgraph of $H_{n,k}$ with clique number $m<n$ is $\left( {\begin{array}{c}3m+1\\ 3\end{array}}\right)$-colourable.

We now consider two cases. If f(n) is finite, we put ${\mathcal {H}}_n=\{H_{n,f(n)}\}$. Otherwise $f(n)=\infty$, and we put ${\mathcal {H}}_n=\{H_{n,k}:k\geqslant n\}$. Finally, we let ${\mathcal {H}}=\bigcup _{n=2}^\infty {\mathcal {H}}_n$, and we let ${\mathcal {G}}$ be the hereditary closure of ${\mathcal {H}}$, i.e., the family of all induced subgraphs of the graphs in ${\mathcal {H}}$.

We now argue that $\chi _{\mathcal {G}}(n)=f(n)$ for all $n\in \mathbb {N}$. The claim holds trivially for $n=1$, so assume $n\geqslant 2$. If $f(n)=\infty$, then the sequence of graphs $\{H_{n,k}:k\geqslant n\}\subseteq {\mathcal {G}}$ all have clique number equal to n and have unbounded chromatic number, thus showing that $\chi _{\mathcal {G}}(n)=\infty$, as claimed. Otherwise, f(n) is finite. The graph $H_{n,f(n)}\in {\mathcal {G}}$ shows that $\chi _{\mathcal {G}}(n)\geqslant f(n)$. For the reverse inequality, let $G\in {\mathcal {G}}$ be such that $\omega (G)=n$. Then there exist integers k and $n^*$ with $k\geqslant n^*\geqslant n$ such that G is an induced subgraph of $H_{n^*,k}\in {\mathcal {H}}$. The unique graph of ${\mathcal {H}}$ with clique number n is $H_{n,f(n)}$. So if $n^*=n$, then $\chi (G)\leqslant \chi (H_{n,f(n)})=f(n)$, and if $n^*>n$, then $\chi (G)\leqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right)$. Combining these inequalities, we conclude that

$$\begin{aligned} f(n) \leqslant \chi _{\mathcal {G}}(n) \leqslant \max \{\genfrac(){0.0pt}1{3n+1}{3},\,f(n)\} = f(n) \text {,} \end{aligned}$$

and the theorem follows. $\square$

The rest of the paper is devoted to proving Theorem 2. We begin with the following lemma.

Lemma 4

For every positive integer k, there is a graph $G_k$ and an acyclic orientation of its edges with the following properties:

• (1) $\chi (G_k)=k$;

• (2) for every pair of vertices u and v, there is at most one directed path from u to v in $G_k$;

• (3) there is a directed path in $G_k$ on k vertices;

• (4) there is a k-colouring $\phi$ of $G_k$ such that $\phi (u)\ne \phi (v)$ for any two distinct vertices u and v such that there is a directed path from u to v in $G_k$.

Various well-known constructions of triangle-free graphs with arbitrarily high chromatic number, such as Zykov’s [13] and Tutte’s [3, 4], satisfy the condition of Lemma 4 once the edges are oriented in a way that follows naturally from the construction. See [2] and [8] for an explicit construction of the graphs $G_k$ with the appropriate acyclic orientations, based on Zykov’s construction. It is only implicit that the acyclic orientations of the graphs in [2] and [8] satisfy all of the properties in the conclusion of Lemma 4, so for the sake completeness we provide a proof based on Tutte’s construction.

Proof of Lemma 4

We proceed by induction on k. The base case $k=1$ follows by taking a single-vertex graph as $G_1$. For the induction step, assume $G_{k-1}$ is an acyclically oriented graph satisfying conditions (1)–(4) for $k-1$. To construct $G_k$, begin with a stable set S with $|S|=(k-1)(|V(G_{k-1})|-1)+1$, and for every subset X of S with $|X|=|V(G_{k-1})|$, add an isomorphic copy $G_X$ of $G_{k-1}$ (with the same orientation as in $G_{k-1}$) and an arbitrary perfect matching between the vertices in X and the vertices of $G_X$, oriented from X to $G_X$. This clearly preserves acyclicity of the orientation. Since the vertices in S have in-degree zero, either every directed path is contained in some copy $G_X$ of $G_{k-1}$, or the starting vertex u is contained in S and every other vertex is contained in some copy $G_X$ of $G_{k-1}$. As every vertex in S has at most one edge to each copy $G_X$ of $G_{k-1}$, the induction hypothesis implies that condition (2) is preserved. Any directed path on $k-1$ vertices in $G_X$ extends to a directed path on k vertices in $G_k$ by adding a vertex from S, so (3) holds. Any colouring of the copies $G_X$ of $G_{k-1}$ with a common palette of $k-1$ colours extends to a k-colouring of $G_k$ by using a single new colour on S, which shows that $\chi (G_k)\leqslant \chi (G_{k-1})+1$ and condition (4) is preserved. Finally, suppose there exists a $(k-1)$-colouring of $G_k$. Then, since $|S|>(k-1)(|V(G_{k-1})|-1)$, there is a monochromatic set $X\subset S$ with $|X|=|V(G_{k-1})|$. Since X and $G_X$ are connected by a perfect matching, at most $k-2$ colours are used on $G_X$, which contradicts the fact that $\chi (G_X)=\chi (G_{k-1})=k-1$. Hence $\chi (G_k)=k$, as claimed in (1). $\square$

For the rest of the argument, we fix an arbitrary sequence $(G_k)_{k\in \mathbb {N}}$ of graphs given by Lemma 4. Now, for every pair of positive integers k and p, where p is a prime number, we construct a graph $G_{k,p}$ by adding edges to $G_k$ as follows.

Let $\leqslant$ be the directed reachability order of the vertices of $G_k$, that is, $u\leqslant v$ if and only if there is a (unique) directed path from u to v in $G_k$. Since the orientation of $G_k$ given by Lemma 4 is acyclic, $\leqslant$ is indeed a partial order. For every pair of vertices u and v in $G_k$ such that $u\leqslant v$, let d(u, v) be the length of the unique directed path from u to v in $G_k$ (i.e., the number of edges in that path). The graph $G_{k,p}$ has the same vertex set as $G_k$ and has the set $\{uv:u<v$ and $d(u,v)\not \equiv 0\pmod {p}\}$ as the edge set. We consider each such edge uv as oriented from u to v. Since the original (oriented) edges uv of $G_k$ satisfy $u<v$ and $d(u,v)=1$, the graph $G_{k,p}$ contains $G_k$ as a subgraph. Furthermore, every edge of $G_{k,p}$ connects vertices with different colours in a k-colouring $\phi$ of $G_k$ claimed in Lemma 4. Therefore, $\chi (G_{k,p})=k$. Furthermore, $G_{k,p}$ is acyclic since $G_k$ is acyclic.

Next, we examine cliques in $G_{k,p}$ (and its induced subgraphs). Since $G_{k,p}$ is acyclic, every clique of $G_{k,p}$ induces a transitive tournament. Given a clique C of an acyclic oriented graph, we let t(C) be the unique in-degree zero vertex of the transitive tournament induced by C. We call t(C) the tail of C. Given a clique C of $G_{k,p}$, we let r(C) be the subset of $\mathbb {Z}_p$ such that $r(C)\equiv \{d(t(C),v):v\in C\}\pmod {p}$. We call r(C) the residue of the clique C. Note that 0 is always contained in r(C) since $t(C)\in C$. Furthermore, $|C|=|r(C)|$, otherwise there would exist two distinct vertices $u,v\in C$ such that $d(t(C),u)\equiv d(t(C),v)\pmod {p}$, and so $d(u,v)\equiv 0\pmod {p}$, which would contradict the fact that u and v are adjacent. This observation allows us to determine the clique number of $G_{k,p}$.

Lemma 5

For every positive integer k and every prime $p\leqslant k$, the graph $G_{k,p}$ has clique number p.

Proof

Since $G_k$ contains a directed path on k vertices and $p\leqslant k$, the graph $G_{k,p}$ contains a clique of size p. Conversely, if C is a clique in $G_{k,p}$, then $|C|=|r(C)|\leqslant |\mathbb {Z}_p|=p$. $\square$

A rotation of a subset X of $\mathbb {Z}_p$ is a subset of $\mathbb {Z}_p$ of the form $X+a=\{x+a:x\in X\}$ for any $a\in \mathbb {Z}_p$. A subset of $\mathbb {Z}_p$ is rooted if it contains 0. The rotation $X+a$ of a rooted subset X of $\mathbb {Z}_p$ is rooted if and only if $-a\in X$. Let $\sim _p$ be the equivalence relation on the rooted subsets of $\mathbb {Z}_p$ such that $X\sim _pY$ whenever Y is a rotation of X. Let $[X]_p$ denote the equivalence class of X in $\sim _p$. For every proper rooted subset X of $\mathbb {Z}_p$ (such that $X\ne \mathbb {Z}_p$), since p is a prime, all rotations $X+a$ of X with $a\in \mathbb {Z}_p$ are distinct. (Indeed, if $X+a=X$, then $\sum _{x\in X}x\equiv \sum _{x \in X}(x+a)\equiv \sum _{x\in X}x+a\cdot |X|\pmod {p}$, so $a\cdot |X|\equiv 0\pmod {p}$, which yields $a\equiv 0\pmod {p}$.) In particular, we have $|[X]_p|=|X|$. Order every equivalence class arbitrarily, and for every proper rooted subset X of $\mathbb {Z}_p$, let $c(X)\in \{1,\ldots ,|X|\}$ denote the position of X in this ordering.

Lemma 6

For every positive integer k, every prime p, and every induced subgraph G of $G_{k,p}$ with clique number $m<p$, we have $\chi (G)\leqslant \left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$.

Proof

We will colour the vertices of G by triples of integers (a, b, c) with $m\geqslant a\geqslant b\geqslant c\geqslant 1$. Since there are $\left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$ choices for such a triple, this will be a $\left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$-colouring of G.

For each vertex v of G, let a(v) be the maximum size of a clique in G with tail v. Thus $m\geqslant a(v)\geqslant 1$. Let B(v) be the intersection of the residues of all cliques of size a(v) with tail v in G. Since 0 belongs to the residue of every clique, we have $0\in B(v)$. Let $b(v)=|B(v)|$, so that $a(v)\geqslant b(v)\geqslant 1$. Let $c(v)=c(B(v))$, so that $b(v)\geqslant c(v)\geqslant 1$, as $|[B(v)]_p|=|B(v)|=b(v)$. Finally, let $\psi (v)=(a(v),b(v),c(v))$. We have $m\geqslant a(v)\geqslant b(v)\geqslant c(v)\geqslant 1$ for every v, so it remains to show that $\psi$ is a proper colouring of G.

Suppose for the sake of contradiction that some two vertices u and v of G with $\psi (u)=\psi (v)$ are connected by an edge of G oriented from u to v. Let $d\in \mathbb {Z}_p$ be such that $d(u,v)\equiv d\pmod {p}$. Since u and v are adjacent in G, we have $d\ne 0$. Observe that if C is a clique in G with residue X and tail v, then prepending u to C and possibly removing the unique vertex w in C with $d(v,w)\equiv -d\pmod {p}$ (if it exists) gives us a clique with residue $(X+d)\cup \{0\}$ and tail u. Therefore, since $a(u)=a(v)$, the residue of every clique of size a(v) with tail v must contain $-d$. Thus $-d\in B(v)$, and if X is the residue of a clique of size a(v) with tail v, then $X+d$ is the residue of a clique of the same size with tail u. Hence $B(u)\subseteq B(v)+d$, and since $b(u)=b(v)$, we further conclude that $B(u)=B(v)+d$. Since 0 belongs to the residue of every clique, both B(u) and B(v) are rooted and $B(u)\sim _pB(v)$. Thus $B(u)=B(v)$, as $c(u)=c(v)$. However, since $b(u)=b(v)\leqslant m<p$ and $d\ne 0$, we have $B(u)=B(v)+d\ne B(v)$, which is a contradiction. This shows that $\psi$ is a proper colouring of G, as desired. $\square$

By combining Lemmas 5 and 6, we have so far proven Theorem 3. Next, we extend the construction to non-primes in order to prove Theorem 2.

For every triple of positive integers k, n, p with p prime and $p>2n$, we construct a subgraph $G_{k,n,p}$ of $G_{k,p}$ by including the edge uv if and only if $d(u,v)\equiv x\pmod {p}$ where $x\in \{\pm 1,\pm 2,\ldots ,\pm (n-1)\}$. We will now determine the clique number and the chromatic number of $G_{k,n,p}$.

Lemma 7

Let k, n, and p be positive integers with p prime, $p>2n$, and $k\geqslant n$. Then $G_{k,n,p}$ has clique number n and chromatic number k.

Proof

We have $\chi (G_k)=\chi (G_{k,p})=k$. Since $G_k$ is a subgraph of $G_{k,n,p}$ and $G_{k,n,p}$ is a subgraph of $G_{k,p}$, it follows that $\chi (G_{k,n,p})=k$. Next, we determine the clique number of $G_{k,n,p}$.

Let $I=\{n,n+1,\ldots ,p-n\}\subset \mathbb {Z}_p$. Note that uv is an edge of $G_{k,n,p}$ if and only if $u<v$ and $d(u,v)\notin \{0\}\cup I\pmod {p}$. Since $G_k$ contains a directed path on k vertices and $n\leqslant k$, the graph $G_{k,n,p}$ has clique number at least n. It remains to show that $G_{k,n,p}$ has clique number at most n.

Let C be a clique in $G_{k,n,p}$, and let $v=t(C)$. If there are vertices $x,y\in C$ with $d(v,y)\in I+d(v,x)\pmod {p}$, then either $y<x$ and $d(y,x)=d(v,x)-d(v,y)=-(d(v,y)-d(v,x))\in -I=I\pmod {p}$, or $x<y$ and $d(x,y)=d(v,y)-d(v,x)\in I\pmod {p}$. In either case, xy is not an edge of $G_{k,n,p}$, contradicting the assumption that C is a clique. Thus the set r(C) is disjoint from the set $\bigcup _{x\in C}(I+d(v,x))=I+r(C)$, which implies that r(C) contains at most one of i and $p-n+i$ for each $i\in \{1,2,\ldots ,n-1\}$. Since $r(C)\subset \{0,1,\ldots ,n-1,p-n+1,p-n+2,\ldots ,p-1\}$, we conclude that $|C|=|r(C)|\leqslant n$. $\square$

Next we examine the maximum size of a clique in an induced subgraph of $G_{k,n,p}$ that is induced by the vertices of a clique in $G_{k,p}$. This will allow us to compare the chromatic number of induced subgraphs of $G_{k,p}$ and $G_{k,n,p}$ that have the same vertex set.

Lemma 8

Let k, n, and p be positive integers with p prime, $p>2n$, and $k\geqslant n$. Then for every clique C of $G_{k,p}$, the induced subgraph $G_{k,n,p}[C]$ of $G_{k,n,p}$ contains a clique of size at least $\frac{n}{p}|C|$.

Proof

Let C be a clique in $G_{k,p}$. For each $i\in \mathbb {Z}_p$, let $J_i=\{i,i+1,\ldots ,i+n-1\}\subset \mathbb {Z}_p$. Since each $i\in \mathbb {Z}_p$ is contained in exactly n of the p sets $J_0,\ldots ,J_{p-1}$, by the pigeon-hole principle, there exists $i\in \mathbb {Z}_p$ such that $|r(C)\cap J_i|\geqslant \frac{n}{p}|r(C)|$. Let $C_i=\{v\in C:d(t(C),v)\in J_i\pmod {p}\}$. It follows that $|C_i|=|r(C)\cap J_i|\geqslant \frac{n}{p}|r(C)|=\frac{n}{p}|C|$. It remains to show that $C_i$ is a clique in $G_{k,n,p}$.

Let x and y be distinct vertices in $C_i$. Since $x,y\in C$, they are adjacent in $G_{k,p}$, so $d(t(C),x)\not \equiv d(t(C),y)\pmod {p}$, and we can assume without loss of generality that $x<y$. It follows that $d(x,y)=d(t(C),y)-d(t(C),x)\in \{\pm 1,\pm 2,\ldots ,\pm (n-1)\}\pmod {p}$, as $d(t(C),x),\,d(t(C),y)\in J_i\pmod {p}$. Hence x and y are adjacent in $G_{k,n,p}$. We conclude that $C_i$ is indeed a clique in $G_{k,n,p}$. $\square$

Lemma 9

Let k, n, and p be positive integers with p prime, $p>2n$, and $k\geqslant n$, and let G be an induced subgraph of $G_{k,n,p}$ with $m=\omega (G)<n$. Then $\chi (G)\leqslant \left( {\begin{array}{c}\lfloor mp/n\rfloor +2\\ 3\end{array}}\right)$.

Proof

Let $G'=G_{k,p}[V(G)]$. Lemma 8 yields $\omega (G')\leqslant \lfloor mp/n\rfloor$. The fact that G is a subgraph of $G'$ and Lemma 6 yield $\chi (G)\leqslant \chi (G')\leqslant \left( {\begin{array}{c}\lfloor mp/n\rfloor +2\\ 3\end{array}}\right)$. $\square$

Theorem 2 now follows from Lemma 7, Lemma 9, and the following theorem of Schur [10] on the gaps between prime numbers.

Theorem 10

For every integer $n\geqslant 2$, there is a prime p such that $2n<p<3n$.

3 Concluding Remarks

To better understand $\chi$-bounding functions, it is of course of interest to improve the bound of $\left( {\begin{array}{c}3m+1\\ 3\end{array}}\right)$ in Theorem 2 (and equivalently this same lower bound function for f in Theorem 1).

A slight tweak to the last step of the proof improves this bound slightly to $\left( {\begin{array}{c}2m\\ 3\end{array}}\right) +o(m^3)$. To do this, instead of using Theorem 10, we can use the fact that for any $\epsilon >0$, there exists a $n_\epsilon$ such that for every $n\geqslant n_\epsilon$, there is always a prime p with $2n<p<(2+\epsilon )n$. This follows from the prime number theorem that the number of primes at most n is asymptotically equal to $n/\ln n$. For a more recent and explicit result on the gaps between primes, see [5].

One may hope that another way to further improve this bound would be to improve the bound of $\left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$ in Lemma 6. However, in our construction, Lemma 6 is in some sense best possible. For every prime p, we have been able to construct a graph $G_k'$ (with k large enough) that satisfies the conclusion of Lemma 4, and such that for every positive integer $m<p$, the graph $G_{k,p}'$ (as constructed from $G_k'$) contains an induced subgraph with clique number m and chromatic number $\left( {\begin{array}{c}m+2\\ 3\end{array}}\right)$. So any improvements would require an entirely new construction.

In the other direction, the only result restricting $\chi$-bounding functions is that of Scott and Seymour [11] stating that if a hereditary class of graphs ${\mathcal {C}}$ satisfies $\chi _{\mathcal {C}}(2)\leqslant 2$, then ${\mathcal {C}}$ is $\chi$-bounded. We conjecture the following generalisation.

Conjecture 11

For every integer $k\geqslant 2$, if ${\mathcal {C}}$ is a hereditary class of graphs such that $\chi _{\mathcal {C}}(n)\leqslant k$ for every positive integer $n\leqslant k$, then the class ${\mathcal {C}}$ is $\chi$-bounded.