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.
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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
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 wellknown 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 wellknown 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 trianglefree 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 Ffree 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
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 kcolouring \(\phi \) 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 wellknown constructions of trianglefree 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 singlevertex graph as \(G_1\). For the induction step, assume \(G_{k1}\) is an acyclically oriented graph satisfying conditions (1)–(4) for \(k1\). To construct \(G_k\), begin with a stable set S with \(S=(k1)(V(G_{k1})1)+1\), and for every subset X of S with \(X=V(G_{k1})\), add an isomorphic copy \(G_X\) of \(G_{k1}\) (with the same orientation as in \(G_{k1}\)) 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 indegree zero, either every directed path is contained in some copy \(G_X\) of \(G_{k1}\), or the starting vertex u is contained in S and every other vertex is contained in some copy \(G_X\) of \(G_{k1}\). As every vertex in S has at most one edge to each copy \(G_X\) of \(G_{k1}\), the induction hypothesis implies that condition (2) is preserved. Any directed path on \(k1\) 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 \(G_{k1}\) with a common palette of \(k1\) colours extends to a kcolouring of \(G_k\) by using a single new colour on S, which shows that \(\chi (G_k)\leqslant \chi (G_{k1})+1\) and condition (4) is preserved. Finally, suppose there exists a \((k1)\)colouring of \(G_k\). Then, since \(S>(k1)(V(G_{k1})1)\), there is a monochromatic set \(X\subset S\) with \(X=V(G_{k1})\). Since X and \(G_X\) are connected by a perfect matching, at most \(k2\) colours are used on \(G_X\), which contradicts the fact that \(\chi (G_X)=\chi (G_{k1})=k1\). 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 kcolouring \(\phi \) 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 indegree 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 nonprimes 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 \(x\in \{\pm 1,\pm 2,\ldots ,\pm (n1)\}\). 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 \(I=\{n,n+1,\ldots ,pn\}\subset \mathbb {Z}_p\). 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 \(pn+i\) for each \(i\in \{1,2,\ldots ,n1\}\). Since \(r(C)\subset \{0,1,\ldots ,n1,pn+1,pn+2,\ldots ,p1\}\), 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 \(J_i=\{i,i+1,\ldots ,i+n1\}\subset \mathbb {Z}_p\). Since each \(i\in \mathbb {Z}_p\) is contained in exactly n of the p sets \(J_0,\ldots ,J_{p1}\), by the pigeonhole 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 \(d(x,y)=d(t(C),y)d(t(C),x)\in \{\pm 1,\pm 2,\ldots ,\pm (n1)\}\pmod {p}\), 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.
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M. Briański: partially supported by the Polish National Science Centre grant (BEETHOVEN; UMO2018/31/G/ST1/03718) B. Walczak: partially supported by the Polish National Science Centre Grant No. 2019/34/E/ST6/00443.
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Briański, M., Davies, J. & Walczak, B. Separating Polynomial \(\chi \)Boundedness from \(\chi \)Boundedness. Combinatorica 44, 1–8 (2024). https://doi.org/10.1007/s00493023000543
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DOI: https://doi.org/10.1007/s00493023000543