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Mixing Circular Colourings

Importance: Low ✭
Author(s): Brewster, Richard C.
Noel, Jonathan A.
Subject: Graph Theory
» Coloring
» » Vertex coloring
Keywords: discrete homotopy
graph colourings
mixing
Recomm. for undergrads: yes
Posted by: Jon Noel
on: September 22nd, 2012
Question   Is $ \mathfrak{M}_c(G) $ always rational?

Given a proper $ k $-colouring $ f $ of a graph $ G $, consider the following 'mixing process:'

    \item choose a vertex $ v\in V(G) $; \item change the colour of $ v $ (if possible) to yield a different $ k $-colouring $ f' $ of $ G $.

A natural problem arises: Can every $ k $-colouring of $ G $ be generated from $ f $ by repeatedly applying this process? If so, we say that $ G $ is $ k $-mixing.

The problem of determining if a graph is $ k $-mixing and several related problems have been studied in a series of recent papers [1,2,4-6]. The authors of [4] provide examples which show that a graph can be $ k $-mixing but not $ k' $-mixing for integers $ k' > k $. For example, given $ m\geq3 $ consider the bipartite graph $ L_m $ which is obtained by deleting a perfect matching from $ K_{m,m} $. It is an easy exercise to show that for $ L_m $ is $ k $-mixing if and only if $ k\geq3 $ and $ k\neq m $. This example motivates the following definition.

Definition   Define the mixing threshold of $ G $ to be $$\mathfrak{M}(G) := \min\{\ell\in\mathbb{N}: G\text{ is }k\text{-mixing whenever } k\geq\ell\}.$$

An analogous definition can be made for circular colouring. Recall, a $ (k,q) $-colouring of a graph is a mapping $ f:V(G)\to \{0,1,\dots,k-1\} $ such that if $ uv\in E(G) $, then $ q\leq |f(u)-f(v)|\leq k-q $. As with ordinary colourings, we say that a graph $ G $ is $ (k,q) $-mixing if all $ (k,q) $-colourings of $ G $ can be generated from a single $ (k,q) $-colouring $ f $ by recolouring one vertex at a time.

Definition   Define the circular mixing threshold of $ G $ to be $$\mathfrak{M}_c(G) := \inf\{\ell\in\mathbb{Q}: G\text{ is }(k,q)\text{-mixing whenever } k/q \geq\ell\}.$$

Several bounds on the circular mixing threshold are obtained in [3], including the following which relates the circular mixing threshold to the mixing threshold.

Theorem  (Brewster and Noel $ [3)</b>&nbsp;&nbsp; $] For every graph $ G $, $$\mathfrak{M}_c(G)\leq\max\left\{\frac{|V(G)|+1}{2}, \mathfrak{M}(G)\right\}.$$

As a corollary, we have the following: if $ |V(G)|\leq 2\mathfrak{M}(G)-1 $, then $ \mathfrak{M}_c(G)\leq \mathfrak{M}(G) $. However, examples in [3] show that the ratio $ \mathfrak{M}_c/\mathfrak{M} $ can be arbitrarily large in general.

Regarding the problem of determining if $ \mathfrak{M}_c $ is rational, it is worth mentioning that there are no known examples of graphs $ G $ for which $ \mathfrak{M}_c(G) $ is not an integer.

Other problems are also given in [3]. One can check that $ \mathfrak{M}_c(K_2) = 2 $ and $ \mathfrak{M}(K_2) = 3 $. However, the only graphs which are known to satisfy $ \mathfrak{M}_c < \mathfrak{M} $ are in some sense related to $ K_2 $, eg. trees and complete bipartite graphs.

Question   Is there a non-bipartite graph $ G $ such that $ \mathfrak{M}_c(G) < \mathfrak{M}(G) $?

Using an example from [4], it is shown in [3] that if $ m\geq2 $ is an integer, then there is a graph $ G $ such that $ \mathfrak{M}_c(G) = \chi(G) = m $ if and only if $ m\neq 3 $. A natural question to ask is whether a similar result holds for the circular chromatic number. Again, certain bipartite graphs are an exception.

Question   Is there a non-bipartite graph $ G $ such that $ \mathfrak{M}_c(G) =\chi_c(G) $?

Also, the example of $ K_2 $ shows that the circular mixing threshold is, in general, not attained. However, the following problem is open.

Question   Is the circular mixing threshold always attained for non-bipartite graphs?

For more precise versions of the last three questions, see [3].

Bibliography

[1] P. Bonsma and L. Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoret. Comput. Sci. 410 (2009), (50): 5215--5226.

[2] P. Bonsma, L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between graph colourings: Computational complexity and possible distances. Electronic Notes in Discrete Mathematics 29 (2007): 463--469.

*[3] R. C. Brewster and J. A. Noel. Mixing Homomorphisms and Extending Circular Colourings. Submitted. pdf.

[4] L. Cereceda, J. van den Heuvel, and M. Johnson. Connectedness of the graph of vertex-colourings. Discrete Math. 308 (2008), (5-6): 913--919.

[5] L. Cereceda, J. van den Heuvel, and M. Johnson. Mixing 3-colourings in bipartite graphs. European J. Combin. 30 (2009), (7): 1593--1606.

[6] L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67 (2011), (1): 69--82.

[7] J. A. Noel. "Jonathan Noel - Mixing Circular Colourings." Webpage.


* indicates original appearance(s) of problem.
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