在双向树中请求集合的干扰有向图着色

Out-colourings of Digraphs

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf out-colouring}. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is ${\cal NP}$-complete. We show that for every choice of positive integers $r,k$ there exists a $k$-strong bipartite tournament which needs at least $r$ colours in every out-colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament $P_7$, every strong semicomplete digraph of minimum…

Colouring Non-Even Digraphs

A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a natural directed analogue of the chromatic number of a graph. A digraph D is called even if for every 0-1-weighting of the edges it contains a directed cycle of even total weight. We show that every non-even digraph has dichromatic number at most 2 a…

Decomposing and colouring some locally semicomplete digraphs

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obtain several applications, among which an answer to a conjecture of Naserasr and the first and third authors: if an oriented graph is such that the out-neighbourhood of every vertex induces a transitive tournament, then one can partition its verte…

A note on the neighbour-distinguishing index of digraphs

In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $γ$ of a digraph $D$ is proper if no two arcs with the same head or with the same tail are assigned the same colour. For each vertex $u$ of $D$, we denote by $S_γ^-(u)$ and $S_γ^+(u)$ the sets of colours that appear on the incoming arcs and on the outgoing arcs of $u$, respectively. An arc colouring $γ$ of $D$ is \emph{neighbour-distinguishing} if, for every two adjacent vertices $u$ and $v$ of $D$, the ordered pairs $(S_γ^-(u),S_γ^+(u))$ and $(S_γ^-(v),S_γ^+(v))$ are dis…

有向图着色与弧连通性

Digraph Colouring and Arc-Connectivity

本文刻画了满足二色数等于最大局部边连通性加一的有向图，推广了无向图类似结论与有向图版 Brooks 定理，并引入 Hajós join 的推广以构造 dicritical digraphs 图族。

Space complexity of list H-colouring: a dichotomy

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is h…