Star Colouring of Bounded Degree Graphs and Regular Graphs

A \(k\)-star colouring of a graph \(G\) is a function \(f:V(G)\to\{0,1,\dots,k-1\}\) such that \(f(u)\neq f(v)\) for every edge \(uv\) of \(G\), and every bicoloured connected subgraph of \(G\) is a star. The star chromatic number of \(G\), \(\chi_s(G)\), is the least integer \(k\) such that \(G\) is \(k\)-star colourable. We prove that \(\chi_s(G)\geq \lceil (d+4)/2\rceil\) for every \(d\)-regular graph \(G\) with \(d\geq 3\). We reveal the structure and properties of even-degree regular graphs \(G\) that attain this lower bound. The structure of such graphs \(G\) is linked with a certain type of Eulerian orientations of \(G\). Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for \(p\geq 2\), a \(2p\)-regular graph \(G\) is \((p+2)\)-star colourable only if \(n:=|V(G)|\) is divisible by \((p+1)(p+2)\). For each \(p\geq 2\) and \(n\) divisible by \((p+1)(p+2)\), we construct a \(2p\)-regular Hamiltonian graph on \(n\) vertices which is \((p+2)\)-star colourable. The problem \(k\)-STAR COLOURABILITY takes a graph \(G\) as input and asks whether \(G\) is \(k\)-star colourable. We prove that 3-STAR COLOURABILITY is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For \(k\geq 3\), \(k\)-STAR COLOURABILITY of bipartite graphs of maximum degree \(k\) is NP-complete, and does not even admit a \(2^{o(n)}\)-time algorithm unless ETH fails.