Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function f : N → N we construct a χ-bounded hereditary class of graphs C with the property that for every integer n ≥ 2 there is a graph in C with clique number at most n and chromatic number at least f (n). In particular, this proves that there are hereditary classes of graphs that are χ-bounded but not polynomially χ-bounded.

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

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively.We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ 1 -norm of the Graver basis is bounded by a function of the maximum ℓ 1 -norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists.Our results yield parameterized algorithms for integer programming when parameterized by the ℓ 1 -norm of the Graver basis of the constraint matrix, when parameterized by the ℓ 1 -norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

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