3 research outputs found
Star Colouring of Bounded Degree Graphs and Regular Graphs
A -star colouring of a graph is a function
such that for every edge of
, and every bicoloured connected subgraph of is a star. The star
chromatic number of , , is the least integer such that is
-star colourable. We prove that for
every -regular graph with . We reveal the structure and
properties of even-degree regular graphs that attain this lower bound. The
structure of such graphs is linked with a certain type of Eulerian
orientations of . 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 , a -regular graph is
-star colourable only if is divisible by . For
each and divisible by , we construct a -regular
Hamiltonian graph on vertices which is -star colourable.
The problem -STAR COLOURABILITY takes a graph as input and asks
whether is -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
, -STAR COLOURABILITY of bipartite graphs of maximum degree is
NP-complete, and does not even admit a -time algorithm unless ETH
fails
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 , a -colouring of a graph is a function
such that for every edge of
. A -colouring of is called a -star colouring of if there is
no path in with and . A -colouring of
is called a -rs colouring of if there is no path in with
. For , the problem -STAR COLOURABILITY
takes a graph as input and asks whether admits a -star colouring.
The problem -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 -star colouring
and -rs colouring with respect to the maximum degree for all . For
, let us denote the least integer such that -STAR COLOURABILITY
(resp. -RS COLOURABILITY) is NP-complete for graphs of maximum degree by
(resp. ).
We prove that for and , -STAR COLOURABILITY is NP-complete
for graphs of maximum degree . We also show that -RS COLOURABILITY is
NP-complete for planar 3-regular graphs of girth 5 and -RS COLOURABILITY is
NP-complete for triangle-free graphs of maximum degree for .
Using these results, we prove the following: (i) for and ,
-STAR COLOURABILITY is NP-complete for -regular graphs if and only if
; and (ii) for , -RS COLOURABILITY is NP-complete
for -regular graphs if and only if