无环、星形和注入着色：H-无图的复杂性图景

Acyclic edge colourings of graphs with large girth

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)Δ$ colours.

Acyclic colourings of graphs with bounded degree

A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper we consider some generalised acyclic $k$-colourings, namely, we require that each colour class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic $5$-colouring such that each colour class induces an acyclic graph with maximum degree at most 4. We prove that the problem of de…

Acyclic, Star, and Injective Colouring: Bounding the Diameter

We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic…

给定单射色数的无限族平面图

Infinite families of planar graphs of a given injective chromatic number

本研究通过计算验证与无限族3-连通平面图构造，拓展了平面图单射着色相关猜想的研究成果，给出多个经典猜想的无限反例并为修正猜想提供支撑。

不含障碍物的图的无环着色

Acyclic colourings of graphs with obstructions

研究了不含特定子图的图的无环着色问题，得出了不同子图类型下的无环色数上界，并通过随机图分析证明了结果的紧性。

Partitioning H-Free Graphs of Bounded Diameter

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by …