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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)

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)

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)

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)

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)

We prove a " supersaturation-type " extension of both Sperner's Theorem (1928) and its generalization by Erd˝ os (1945) to k-chains. Our result implies that a largest family whose size is x more than the size of a largest k-chain free family and that contains the minimum number of k-chains is the family formed by taking the middle (k − 1) rows of the… (More)