Related papers: Van der Waerden's Colouring Theorem and Classical …
In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem,…
The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…
Despite their ionic nature, many layered diatomic crystals avoid internal electric polarization by forming a centrosymmetric lattice at their optimal anti-parallel van-der-Waals stacking. Here, we report a stable ferroelectric order…
Coloring planar Feynman diagrams in spinor quantum electrodynamics, is a non trivial model soluble without computer. Four colors are necessary and sufficient.
The generalized decoration-iteration transformation is adopted to treat exactly a hybrid model of doubly decorated two-dimensional lattices, which have localized Ising spins at their nodal lattice sites and itinerant electrons delocalized…
The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an…
We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…
We study classical Ising spin-$\frac{1}{2}$ models on the 2D square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin…
We propose color magnetism as a generalization of the ordinary Heisenberg (anti-)ferro magnets on a triangular lattice. Vortex matter consisting of an Abrikosov lattice of non-Abelian vortices with color magnetic fluxes shows a color ferro…
It was argued many years ago that translational symmetry breaking due to the appearance of spin-Peierls ordering (or bond-charge stripe order) is a fundamental property of the quantum paramagnetic states of a large class of square lattice…
A spin-1 Ising model incorporating positional order to a standard lattice gas with no attractive interactions is introduced and found to be consistent with all known attributes of the freezing transition of the hard-sphere system.…
A vertex coloring of a simplicial complex $\Delta$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $\Delta$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that…
We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its…
We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite…
We define "paradoxical colouring rule", show its relation to measure theoretic paradoxes, and demonstrate that proper vertex colouring can be a paradoxical colouring rule.
Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this…
Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between \emph{optimization} and…
Some results about phase separation in coupled map lattices satisfying a conservation law are presented. It is shown that this constraint is the origin of interesting antiferromagnetic effective couplings and allows transitions to…
A new set of exact scattering matrices in 1+1 dimensions is proposed by solving the bootstrap equations. Extending earlier constructions of colour valued scattering matrices this new set has its colour structure associated to non…
An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements…