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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)

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)

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)

Given a graph G = (V, E), let P be a partition of V. We say that P is dominating if, for each part P of P, the set V \ P is a dominating set in G (equivalently, if every vertex has a neighbour of a different colour from its own). We say that P is acyclic if for any parts P, P of P, the bipartite subgraph G[P, P ] consisting of the edges between P and P in P… (More)

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.