Modulo-Counting First-Order Logic on Bounded Expansion Classes

We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic lift, to an existential first-order formula. As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes. As an application, we prove that a class has structurally bounded expansion if and only if is a class of bounded depth vertex-minors of graphs in a bounded expansion class. We also show how our results can be used to implement fast matrix calculus on bounded expansion matrices over a finite field.Comment: submitted to CSGT2022 special issu

Subchromatic numbers of powers of graphs with excluded minors

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $\chi_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$ admits a $k$-subcolouring. Ne\v{s}et\v{r}il, Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for $\chi_{\textrm{sub}}(G^2)$ when $G$ is planar. We show that $\chi_{\textrm{sub}}(G^2)\le 43$ 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 $\chi_{\textrm{sub}}(G^{3})\le 95$, 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 $\chi_{\textrm{sub}}(G^p)$ for all $p$, 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 coming from any fixed class with bounded layered cliquewidth. In particular, this implies a 2-approximation algorithm for the subchromatic number of powers $G^p$ of graphs coming from any fixed class with bounded layered treewidth (such as the class of planar graphs). This algorithm works even if the power $p$ and the graph $G$ is unknown.Comment: 21 pages, 2 figure

Improper colourings inspired by Hadwiger’s conjecture

Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree

Stretchability Of Jordan Arc Contact Systems

We prove that a necessary and sufficient condition for a contact system of Jordan arcs to be stretchable is to be extendable into a weak arrangement of pseudo-lines