@article{19002, abstract = {A k-subcolouring of a graph G is a function f : V (G) → {0,...,k − 1} such that the set of vertices coloured i induce a disjoint union of cliques. The subchromatic number, χsub(G), is the minimum k such that G admits a k-subcolouring. Nešetril, ˇ Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for χsub(G2) when G is planar. We show that χsub(G2) ≤ 43 when G is planar, improving their bound of 135. We give even better bounds when the planar graph G has larger girth. Moreover, we show that χsub(G3) ≤ 95, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs. We give improved bounds for χsub(Gp) for all p ≥ 2, whenever G has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes. Finally, we give a 2-approximation algorithm for the subchromatic number of graphs having a layering in which each layer has bounded cliquewidth and this layering is computable in polynomial time (like the class of all dth powers of planar graphs, for fixed d). This algorithm works even if the power p and the graph G is unknown.}, author = {Cortés, Pedro P. and Kumar, Pankaj and Moore, Benjamin and Ossona de Mendez, Patrice and Quiroz, Daniel A.}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {4}, publisher = {Elsevier}, title = {{Subchromatic numbers of powers of graphs with excluded minors}}, doi = {10.1016/j.disc.2024.114377}, volume = {348}, year = {2025}, } @article{22163, abstract = {For a field F and integers d and k, a set A ⊆ Fd is called k-nearly orthogonal if its members are non-self-orthogonal and every k + 1 vectors of A include an orthogonal pair. We prove that for every prime p there exists some δ = δ(p)> 0, such that for every field F of characteristic p and for all integers k ≥ 2 and d ≥ k, there exists a k-nearly orthogonal set of at least dδ·k/ logk vectors of Fd. The size of the set is optimal up to the logk term in the exponent. We further prove two extensions of this result. In the first, we provide a large set A of non-self-orthogonal vectors of Fd such that for every two subsets of A of size k+1 each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every k + 1 vectors of the produced set A include ℓ + 1 pairwise orthogonal vectors for an arbitrary fixed integer 1 ≤ ℓ ≤ k. The proofs involve probabilistic and spectral arguments and the hypergraph container method}, author = {Haviv, Ishay and Mattheus, Sam and Milojević, Aleksa and Wigderson, Yuval}, issn = {0012-365X}, journal = {Discrete Mathematics}, keywords = {Nearly orthogonal sets, Ramsey theory, Finite fields}, number = {4}, publisher = {Elsevier}, title = {{Larger nearly orthogonal sets over finite fields}}, doi = {10.1016/j.disc.2024.114373}, volume = {348}, year = {2025}, } @article{15163, abstract = {For some k∈Z≥0∪{∞}, we call a linear forest k-bounded if each of its components has at most k edges. We will say a (k,ℓ)-bounded linear forest decomposition of a graph G is a partition of E(G) into the edge sets of two linear forests Fk,Fℓ where Fk is k-bounded and Fℓ is ℓ-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both k and ℓ are at least 2, NP-complete if k≥9 and ℓ=1, and is in P for (k,ℓ)=(2,1). Before this, the only known NP-complete cases were the (2,2) and (3,3) cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than 3-edge-colouring such graphs.}, author = {Campbell, Rutger and Hörsch, Florian and Moore, Benjamin}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {6}, publisher = {Elsevier}, title = {{Decompositions into two linear forests of bounded lengths}}, doi = {10.1016/j.disc.2024.113962}, volume = {347}, year = {2024}, } @article{12680, abstract = {The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines.}, author = {Ivanov, Grigory and Köse, Seyda}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {6}, publisher = {Elsevier}, title = {{Erdős-Ko-Rado and Hilton-Milner theorems for two-forms}}, doi = {10.1016/j.disc.2023.113363}, volume = {346}, year = {2023}, } @article{9098, abstract = {We study properties of the volume of projections of the n-dimensional cross-polytope $\crosp^n = \{ x \in \R^n \mid |x_1| + \dots + |x_n| \leqslant 1\}.$ We prove that the projection of $\crosp^n$ onto a k-dimensional coordinate subspace has the maximum possible volume for k=2 and for k=3. We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of $\crosp^n$ onto a k-dimensional subspace for any n>k⩾2.}, author = {Ivanov, Grigory}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {5}, publisher = {Elsevier}, title = {{On the volume of projections of the cross-polytope}}, doi = {10.1016/j.disc.2021.112312}, volume = {344}, year = {2021}, } @article{6638, abstract = {The crossing number of a graph G is the least number of crossings over all possible drawings of G. We present a structural characterization of graphs with crossing number one.}, author = {Silva, André and Arroyo Guevara, Alan M and Richter, Bruce and Lee, Orlando}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {11}, pages = {3201--3207}, publisher = {Elsevier}, title = {{Graphs with at most one crossing}}, doi = {10.1016/j.disc.2019.06.031}, volume = {342}, year = {2019}, } @article{4065, abstract = {We prove that given n⩾3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n−9 sides, and with not more than 3n−6 distinct slopes. Furthermore, we construct sets that require 6n−9 sides and 3n−6 slopes for n⩾3. The upper bound on the number of slopes implies a new bound on a recently studied transversal problem.}, author = {Edelsbrunner, Herbert and Robison, Arch and Shen, Xiao}, issn = {1872-681X}, journal = {Discrete Mathematics}, number = {2}, pages = {153 -- 164}, publisher = {Elsevier}, title = {{Covering convex sets with non-overlapping polygons}}, doi = {10.1016/0012-365X(90)90147-A}, volume = {81}, year = {1990}, } @article{4107, abstract = {A set of m planes dissects E3 into cells, facets, edges and vertices. Letting deg(c) be the number of facets that bound a cellc, we give exact and asymptotic bounds on the maximum of ∈cinCdeg(c), if C is a family of cells of the arrangement with fixed cardinality.}, author = {Edelsbrunner, Herbert and Haussler, David}, issn = {1872-681X}, journal = {Discrete Mathematics}, number = {C}, pages = {139 -- 146}, publisher = {Elsevier}, title = {{The complexity of cells in 3-dimensional arrangements}}, doi = {10.1016/0012-365X(86)90008-7}, volume = {60}, year = {1986}, }