The Complexity of Star Colouring in Bounded Degree Graphs and Regular Graphs

Abstract

A k-star colouring of a graph G is a function $f:V(G)\rightarrow \{0,1,\dots ,k-1\}$ such that $f(u)\ne f(v)$ for every edge uv of G, and G does not contain a 4-vertex path bicoloured by f as a subgraph. For $k\in \mathbb {N}$, the problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. By the construction of Coleman and Moré (SIAM J. Numer. Anal., 1983), for all $k\ge 3$, k-Star Colourability is NP-complete for graphs of maximum degree $d=k(k-1+\lceil \sqrt{k} \rceil )$. For $k=4$ and $k=5$, the maximum degree in this NP-completeness result is $d=20$ and $d=35$ respectively. We reduce the maximum degree to $d=4$ in both cases: i.e., 4-Star Colourability and 5-Star Colourability are NP-complete for graphs of maximum degree four. We also show that for all $k\ge 3$ and $d<k$, the time complexity of k-Star Colourability is the same for graphs of maximum degree d and d-regular graphs (i.e., the problem is either in P for both classes or NP-complete for both classes).

M. A. Shalu—Supported by SERB(DST), MATRICS scheme MTR/2018/000086.