Colouring graphs when the number of colours is nearly the maximum degree

ABSTRACT

We consider for graphs of maximum degree &Dgr;, the problem of determining whether &khgr;G) > &Dgr;-k for various values of k. We obtain sharp theorems characterizing when the barrier to &Dgr;-k colourability must be a local condition, i.e. a small subgraph, and when it can be global. We also show that for large fixed &Dgr;, this problem is either NP-complete or can be solved in linear time, and we determine precisely which values of k correspond to each case prove that Hitting Set with sets of size B is hard to approximate to within a factor $B^{1/19}$. The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B=2. The relationship between hardness of approximation and set size seems to have not been explored before.

REFERENCES

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

Additional Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES
      F.1.2 Modes of Computation
          Subjects: Probabilistic computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

General Terms:
Algorithms, Performance, Theory, Verification

Keywords:
graph colouring, probabilistic methods