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

Uniquely circular colourable and uniquely fractional colourable graphs of large girth

Given any rational numbers $r \geq r' >2$ and an integer $g$, we prove that there is a graph $G$ of girth at least $g$, which is uniquely $r$-colourable and uniquely $r'$-fractional colourable

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

Graph Partitioning With Input Restrictions

In this thesis we study the computational complexity of a number of graph partitioning problems under a variety of input restrictions. Predominantly, we research problems related to Colouring in the case where the input is limited to hereditary graph classes, graphs of bounded diameter or some combination of the two. In Chapter 2 we demonstrate the dramatic eect that restricting our input to hereditary graph classes can have on the complexity of a decision problem. To do this, we show extreme jumps in the complexity of three problems related to graph colouring between the class of all graphs and every other hereditary graph class. We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be H-free for some graph H if it contains no induced subgraph isomorphic to H. Similarly, G is said to be H-free for some set of graphs H, if it does not contain any graph in H as an induced subgraph. Here, the set H consists usually of a single cycle or tree but may also contain a number of cycles, for example we give results for graphs of bounded diameter and girth. Chapter 4 is dedicated to three variants of the Colouring problem, Acyclic Colouring, Star Colouring, and Injective Colouring. We give complete or almost complete dichotomies for each of these decision problems restricted to H-free graphs. In Chapter 5 we study these problems, along with three further variants of 3-Colouring, Independent Odd Cycle Transversal, Independent Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of bounded diameter. Finally, Chapter 6 deals with a dierent variety of problems. We study the problems Disjoint Paths and Disjoint Connected Subgraphs for H-free graphs