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.
Similar content being viewed by others
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.
Data Availability:
There is no dataset associated with this manuscript.
References
Briański, M., Davies, J., Walczak, B.: Separating polynomial \(\chi \)boundedness from \(\chi \)boundedness, arXiv:2201.08814v1, (2022)
Carbonero, A., Hompe, P., Moore, B., Spirkl, S.: A counterexample to a conjecture about trianglefree induced subgraphs of graphs with large chromatic number. J. Combinatorial Theor. Ser. B 158, 63–69 (2023)
Descartes, B.: A three colour problem. Eureka 9(21), 24–25 (1947)
Descartes, B.: Solution to advanced problem no. 4526. Am. Math. Mon. 61, 352 (1954)
Dusart, P.: Explicit estimates of some functions over primes. Ramanujan J. 45, 227–251 (2018)
Esperet, L.: Graph colorings, flows and perfect matchings, Habilitation thesis, Université Grenoble Alpes, (2017)
Girão, A., Illingworth, F., Powierski, E., Savery, M., Scott, A., Tamitegama, Y., Tan, J.: Induced subgraphs of induced subgraphs of large chromatic number, arXiv:2203.03612, (2022)
Kierstead, H.A., Trotter, W.T.: Colorful induced subgraphs. Discret. Math. 101, 165–169 (1992)
Schiermeyer, I., Randerath, B.: Polynomial \(\chi \)binding functions and forbidden induced subgraphs: a survey. Graphs Combinatorics 35(1), 1–31 (2019)
Schur, I.: Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, I (Some theorems about prime numbers with applications to irreducibility questions, I). Sitzungsberichte der Preussischen Akademie der Wissenschaften, PhysikalischMathematische Klasse 14, 125–136 (1929)
Scott, A., Seymour, P.: Induced subgraphs of graphs with large chromatic number. I. Odd holes. J. Combinatorial Theor. Ser. B 121, 68–84 (2016)
Scott, A., Seymour, P.: A survey of \(\chi \)boundedness. J. Graph Theor. 95(3), 473–504 (2020)
Zykov, A.A.: O nekotorykh svoystvakh lineynykh kompleksov (On some properties of linear complexes). Matematicheskii Sbornik 24(66), 163–188 (1949)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Briański, M., Davies, J. & Walczak, B. Separating Polynomial \(\chi \)Boundedness from \(\chi \)Boundedness. Combinatorica 44, 1–8 (2024). https://doi.org/10.1007/s00493023000543
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493023000543