Bounding clique-width via perfect graphs

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no subgraph isomorphic to H1 or H2. We continue a recent study into the clique-width of (H1,H2)-free graphs and present three new classes of (H1,H2)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring problem restricted to (H1,H2)-free graphs. The three new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of their clique-width we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs

3-Colourability of Dually Chordal Graphs in Linear Time

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four

Colouring exact distance graphs of chordal graphs

For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$. Recently, there has been an effort to obtain bounds on the chromatic number $\chi(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has been shown that $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ for even $p$. We show that if $G$ is chordal and has tree-width $t$, then $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G))$ for even $p$. If we could show that for every graph $H$ of tree-width $t$ there is a chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width $t$. While we cannot do this, we show that for every graph $H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and contains $H$ as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which arise from reviewers' comment

On embeddings of CAT(0) cube complexes into products of trees

We prove that the contact graph of a 2-dimensional CAT(0) cube complex ${\bf X}$ of maximum degree $\Delta$ can be coloured with at most $\epsilon(\Delta)=M\Delta^{26}$ colours, for a fixed constant $M$. This implies that ${\bf X}$ (and the associated median graph) isometrically embeds in the Cartesian product of at most $\epsilon(\Delta)$ trees, and that the event structure whose domain is ${\bf X}$ admits a nice labeling with $\epsilon(\Delta)$ labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the computation of the bounds in Theorem 1. Some figures repaire

Random Perfect Graphs

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number $\alpha(P_n)$ and clique number $\omega(P_n)$ is close to a concentrated distribution $L(n)$ which plays an important role in our generation method. We also prove that the probability that $P_n$ contains any given graph $H$ as an induced subgraph is asymptotically $0$ or $\frac12$ or $1$. Further we show that almost all perfect graphs are $2$-clique-colourable, improving a result of Bacs\'o et al from 2004; they are almost all Hamiltonian; they almost all have connectivity $\kappa(P_n)$ equal to their minimum degree; they are almost all in class one (edge-colourable using $\Delta$ colours, where $\Delta$ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon $W_P(x, y) = \frac12(\mathbb{1}[x \le 1/2] + \mathbb{1}[y \le 1/2])$