On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor

The generalised colouring numbers $\mathrm{col}_r(G)$ and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the $r$-colouring number $\mathrm{col}_r$ and a polynomial bound for the weak $r$-colouring number $\mathrm{wcol}_r$. In particular, we show that if $G$ excludes $K_t$ as a minor, for some fixed $t\ge4$, then $\mathrm{col}_r(G)\le\binom{t-1}{2}\,(2r+1)$ and $\mathrm{wcol}_r(G)\le\binom{r+t-2}{t-2}\cdot(t-3)(2r+1)\in\mathcal{O}(r^{\,t-1})$. In the case of graphs $G$ of bounded genus $g$, we improve the bounds to $\mathrm{col}_r(G)\le(2g+3)(2r+1)$ (and even $\mathrm{col}_r(G)\le5r+1$ if $g=0$, i.e. if $G$ is planar) and $\mathrm{wcol}_r(G)\le\Bigl(2g+\binom{r+2}{2}\Bigr)\,(2r+1)$.Comment: 21 pages, to appear in European Journal of Combinatoric

On the generalised colouring numbers of graphs that exclude a fixed minor

The generalised colouring numbers colr(G) and wcolr(G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r-colouring number colr and a polynomial bound for the weak r-colouring number wcolr. In particular, we show that if G excludes Kt as a minor, for some fixed t≥4, then colr(G)≤(t−12)(2r+1) and wcolr(G)≤(r+t−2t−2)⋅(t−3)(2r+1)∈O(rt−1). In the case of graphs G of bounded genus g, we improve the bounds to colr(G)≤(2g+3)(2r+1) (and even colr(G)≤5r+1 if g=0, i.e. if G is planar) and wcolr(G)≤(2g+(r+22))(2r+1)

The Generalised Colouring Numbers on Classes of Bounded Expansion

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors

Nowhere Dense Graph Classes and Dimension

Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every $h \geq 1$ and every $\epsilon > 0$, posets of height at most $h$ with $n$ elements and whose cover graphs are in the class have dimension $\mathcal{O}(n^{\epsilon})$.Comment: v4: Minor changes suggested by a refere

Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic

Chromatic numbers of exact distance graphs

For any graph G = (V;E) and positive integer p, the exact distance-p graph G[\p] is the graph with vertex set V , which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[\p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[\p] is bounded by the weak (2

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