The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs P1, . . . ,Pk (usually describing some common graph properties), is to decide, for a given graph G, whether the vertex set of G can be partitioned into sets V1, . . . , Vk such that, for each i, the induced subgraph of G on Vi belongs to Pi. It can be seen that GCOL generalizes many natural colouri...

In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G2 of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a dually chordal graph, then there exists a spanning tree T of G that is an additive 3-spanner as wel...

Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let dmin(G) (dmax(G)) denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of G. We call G fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying dmin(G) 6 d 6 dmax(G). A graph G is c...

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined inO(1) tim...

Exterior algebraic shifting and symmetric algebraic shifting of bipartite graphs and chordal graphs are studied in the present paper. keywords : Algebraic shifting, chordal graph.

A k-colouring of a graph G = (V,E) is a mapping c : V → {1, 2, . . . , k} such that c(u) 6= c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We...

1 I n t r o d u c t i o n A subset D C V is a dominating set in graph G = (V, E) iff for all vertices v E V \ D there is a vertex u E D with uv E E. It is a dominating clique in G iff D is a dominating set in G and a clique (i.e. for all u, v E D uv E E). There are many papers investigating the problem of finding minimum dominating sets in graphs with (and without) additional requirements to th...