The complexity of colouring problems on dense graphs
Theoretical Computer Science
Hard enumeration problems in geometry and combinatorics
SIAM Journal on Algebraic and Discrete Methods
On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
The complexity of some graph colouring problems
Discrete Applied Mathematics
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
An algorithm for the Tutte polynomials of graphs of bounded treewidth
Discrete Mathematics
The complexity of counting colourings and independent sets in sparse graphs and hypergraphs
Computational Complexity
Chromatic polynomials and their zeros and asymptotic limits for families of graphs
Discrete Mathematics - Special issue on the 17th british combinatorial conference selected papers
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width
Combinatorics, Probability and Computing
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It is well known that counting $\lambda$-colourings ($\lambda\geq 3$) is #P-complete for general graphs, and also for several restricted classes such as bipartite planar graphs. On the other hand, it is known to be polynomial time computable for graphs of bounded tree-width. There is often special interest in counting colourings of square grids, and such graphs can be regarded as borderline graphs of unbounded tree-width in a specific sense. We are thus motivated to consider the complexity of counting colourings of subgraphs of the square grid. We show that the problem is #P-complete when $\lambda\geq 3$. It remains #P-complete when restricted to induced subgraphs with maximum degree 3.