Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min...

For every integer r > 3 and every ε > 0 we construct a graph with maximum degree r − 1 whose circular total chromatic number is in the interval (r, r + ε). This proves that (i) every integer r > 3 is an accumulation point of the set of circular total chromatic numbers of graphs, and (ii) for every ∆ > 2, the set of circular total chromatic numbers of graphs with maximum degree ∆ is infinite. Al...

The packing chromatic number χρ(G) of a graphG is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where each Vi is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that ω(G) = a, χ(G) = b, and χρ(G) = c. If so, we say that (a, b, c) is realizable. It is proved that b =...

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible. Consider a graph G where each edge of G is assigned a colour from {1, ..., k} (this is not necessarily a proper edge colouring). A k-adapted colouring is an assignment of colours from {1, ..., k} to the vertices of G such that there is no edge with th...

A flow graph G = (V,E, s) is a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, ci...

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w ∈ Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of thi...

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). In this paper, the method of Alon and Tarsi is used to prove that all multicircuits of even order, and some regular and near-regular multicircuits of odd order have total choosability (i.e., list total chromatic number) equal to their ordinary total chromatic number. This completes the proof...

We say that u is dominated by v if and only if every path from r to u passes through v. It is not difficult to show that if u is dominated by both v1 and v2, then either v1 dominates v2, or v2 dominates v1. We say that v is the immediate dominator of u if and only if every dominator of u is also a dominator of v. It follows from the previous observation that every vertex u has a unique immediat...