# Ross J. Kang

• Electronic Notes in Discrete Mathematics
• 2005
We investigate the following problem proposed by Alcatel. A satellite sends information to receivers on earth, each of which is listening on a chosen frequency. Technically, it is impossible for the satellite to precisely focus its signal onto a receiver. Part of the signal will be spread in an area around its destination and this creates noise for nearby(More)
• 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 &#8712; 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 &#8712; E(G) if and only if(More)
We prove a “supersaturation-type” extension of both Sperner’s Theorem (1928) and its generalization by Erdős (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 themiddle (k−1) rows of the Boolean(More)
• Journal of Graph Theory
• 2009
The circular chromatic number of a graph is defined as follows. A t-circular colouring of a graph G is a map c : V (G) → S(t), where S(t) denotes a circle of circumference t, such that the length of the arc between c(v) and c(w) is at least one whenever vw ∈ E(G). The circular chromatic number χc(G) of G is the infimum of all t for which there is a(More)
• Combinatorics, Probability & Computing
• 2014
We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2− ε)∆t for graphs of maximum degree at most ∆, where ε is some absolute(More)
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• Electronic Notes in Discrete Mathematics
• 2007
We consider the t-improper chromatic number of the Erdős-Rényi random graph Gn,p. The t-improper chromatic number χ(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 probability p(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(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)
• Discrete Mathematics
• 2008
For any graph G, the k-improper chromatic number χ(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 kimproper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of McDiarmid and Reed (1999);(More)