Hardness Transitions of Star Colouring and Restricted Star Colouring

We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For $k\in \mathbb{N}$, a $k$-colouring of a graph $G$ is a function $f\colon V(G)\to \mathbb{Z}_k$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$. A $k$-colouring of $G$ is called a $k$-star colouring of $G$ if there is no path $u,v,w,x$ in $G$ with $f(u)=f(w)$ and $f(v)=f(x)$. A $k$-colouring of $G$ is called a $k$-rs colouring of $G$ if there is no path $u,v,w$ in $G$ with $f(v)>f(u)=f(w)$. For $k\in \mathbb{N}$, the problem $k$-STAR COLOURABILITY takes a graph $G$ as input and asks whether $G$ admits a $k$-star colouring. The problem $k$-RS COLOURABILITY is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of $k$-star colouring and $k$-rs colouring with respect to the maximum degree for all $k\geq 3$. For $k\geq 3$, let us denote the least integer $d$ such that $k$-STAR COLOURABILITY (resp. $k$-RS COLOURABILITY) is NP-complete for graphs of maximum degree $d$ by $L_s^{(k)}$ (resp. $L_{rs}^{(k)}$). We prove that for $k=5$ and $k\geq 7$, $k$-STAR COLOURABILITY is NP-complete for graphs of maximum degree $k-1$. We also show that $4$-RS COLOURABILITY is NP-complete for planar 3-regular graphs of girth 5 and $k$-RS COLOURABILITY is NP-complete for triangle-free graphs of maximum degree $k-1$ for $k\geq 5$. Using these results, we prove the following: (i) for $k\geq 4$ and $d\leq k-1$, $k$-STAR COLOURABILITY is NP-complete for $d$-regular graphs if and only if $d\geq L_s^{(k)}$; and (ii) for $k\geq 4$, $k$-RS COLOURABILITY is NP-complete for $d$-regular graphs if and only if $L_{rs}^{(k)}\leq d\leq k-1$

3-Colourability of Dually Chordal Graphs in Linear Time

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four

On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

A Strategy for Dynamic Programs: Start over and Muddle through

In the setting of DynFO, dynamic programs update the stored result of a query whenever the underlying data changes. This update is expressed in terms of first-order logic. We introduce a strategy for constructing dynamic programs that utilises periodic computation of auxiliary data from scratch and the ability to maintain a query for a limited number of change steps. We show that if some program can maintain a query for log n change steps after an AC$^1$-computable initialisation, it can be maintained by a first-order dynamic program as well, i.e., in DynFO. As an application, it is shown that decision and optimisation problems defined by monadic second-order (MSO) formulas are in DynFO, if only change sequences that produce graphs of bounded treewidth are allowed. To establish this result, a Feferman-Vaught-type composition theorem for MSO is established that might be useful in its own right

On the Complexity of Role Colouring Planar Graphs, Trees and Cographs

We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with $1< k <n$ colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as $k$ is either constant or has a constant difference with $n$, the number of vertices in the tree. Finally, we prove that cographs are always $k$-role-colourable for $1<k\leq n$ and construct such a colouring in polynomial time