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- Frédéric Havet, Ross J. Kang, Jean-Sébastien Sereni
- Electronic Notes in Discrete Mathematics
- 2005

- Ross J. Kang, Tobias Müller
- Discrete & Computational Geometry
- 2011

A graph G is a k-sphere graph if there are k-dimensional real vectors v<sub>1</sub>,..., v<sub>n</sub> such that ij ∈ E(G) if and only if the distance between v<sub>i</sub> and v<sub>j</sub> is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v<sub>1</sub>,...,v<sub>n</sub> such that ij ∈ E(G) if and only if… (More)

- Frédéric Havet, Ross J. Kang, Tobias Müller, Jean-Sébastien Sereni
- Journal of Graph Theory
- 2009

Motivated by a satellite communications problem, we consider a generalised colouring problem on unit disk graphs. A colouring is k-improper if no vertex receives the same colour as k+1 of its neighbours. The k-improper chromatic number χ k (G) is the least number of colours needed in a k-improper colouring of a graph G. The main subject of this work is… (More)

- Ross J. Kang, Colin McDiarmid
- Electronic Notes in Discrete Mathematics
- 2007

We consider the t-improper chromatic number of the Erd˝ os-Rényi random graph G n,p. The t-improper chromatic number χ t (G) of G is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge… (More)

- Ross J. Kang, Tobias Müller, Jean-Sébastien Sereni
- Discrete Mathematics
- 2008

For any graph G, the k-improper chromatic number χ k (G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate the ratio of the k-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of McDiarmid and Reed… (More)

For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic and each colour class induces a graph with maximum degree at most t. In the first part, we show that all subcubic graphs are acyclically 1-improperly 3-choosable,… (More)

- Ross J. Kang, Matthias Mnich, Tobias Müller
- ESA
- 2010

We present a linear-time algorithm that, given a planar graph with m edges and maximum degree 3, finds an induced matching of size at least m/9. This is best possible.

- Ross J. Kang, László Lovász, Tobias Müller, Edward R. Scheinerman
- Electr. J. Comb.
- 2010

A graph G is a k-dot product graph if there exists a vector labelling u : V (G) → R k such that u(i) T u(j) ≥ 1 if and only if ij ∈ E(G). Fiduccia, Scheinerman, Trenk and Zito [4] asked whether every planar graph is a 3-dot product graph. We show that the answer is " no ". On the other hand, every planar graph is a 4-dot product graph. We also answer the… (More)

- Nikolaos Fountoulakis, Ross J. Kang, Colin McDiarmid
- Electr. J. Comb.
- 2010

Given a graph G = (V, E), a vertex subset S ⊆ V is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number α t (G) of G is the maximum order of a t-stable set in G. The theme of this paper is the typical values that this parameter takes on a random graph on n vertices and edge probability equal… (More)