25 research outputs found
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
Schrijver graphs and projective quadrangulations
In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the
authors have extended the concept of quadrangulation of a surface to higher
dimension, and showed that every quadrangulation of the -dimensional
projective space is at least -chromatic, unless it is bipartite.
They conjectured that for any integers and , the
Schrijver graph contains a spanning subgraph which is a
quadrangulation of . The purpose of this paper is to prove the
conjecture
Local chromatic number of quadrangulations of surfaces
The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four.
Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces
replacing the condition of the graph being not bipartite by a more technical condition of
an odd quadrangulation. This paper investigates when these general results are true for the
local chromatic number instead of the chromatic number. Surprisingly, we find out that
(unlike in the case of the chromatic number) this depends on the genus of the surface. For
the non-orientable surfaces of genus at most four, the local chromatic number of any odd
quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5
or higher.
We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of
arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for
the usual chromatic number
Reconfiguring graph homomorphisms on the sphere
Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs H (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle. From this result, we deduce an analogous statement for non-bipartite K2,3-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and 4-chromatic generalized Mycielski graphs, for which it was not.
If we instead consider graphs G and H with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which H-Recolouring is known to be PSPACE-complete for reflexive instances