100 research outputs found
A counterexample to a conjecture on facial unique-maximal colorings
A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face α the maximal color appears exactly once on the vertices of α. Fabrici and Göring [4] proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color theorem. Wendland [6] later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. s we conclude that facial unique-maximum chromatic number of the sphere is five
Four-Color Coloring of a Partial Map of Europe
The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary other than a single point do not share the same color. In this article we attempt to color a partial map of Europe with four color using Artificial Intelligence techniques, defining it as a Constraint Satisfaction Problem (CSP). The algorithm created was succeeded to find all four solutions of the problem
The Eternal Game Chromatic Number of a Graph
Game coloring is a well-studied two-player game in which each player properly
colors one vertex of a graph at a time until all the vertices are colored. An
`eternal' version of game coloring is introduced in this paper in which the
vertices are colored and re-colored from a color set over a sequence of rounds.
In a given round, each vertex is colored, or re-colored, once, so that a proper
coloring is maintained. Player 1 wants to maintain a proper coloring forever,
while player 2 wants to force the coloring process to fail. The eternal game
chromatic number of a graph is defined to be the minimum number of colors
needed in the color set so that player 1 can always win the game on . We
consider several variations of this new game and show its behavior on some
elementary classes of graphs
Geometry of polycrystals and microstructure
We investigate the geometry of polycrystals, showing that for polycrystals
formed of convex grains the interior grains are polyhedral, while for
polycrystals with general grain geometry the set of triple points is small.
Then we investigate possible martensitic morphologies resulting from intergrain
contact. For cubic-to-tetragonal transformations we show that homogeneous
zero-energy microstructures matching a pure dilatation on a grain boundary
necessarily involve more than four deformation gradients. We discuss the
relevance of this result for observations of microstructures involving second
and third-order laminates in various materials. Finally we consider the more
specialized situation of bicrystals formed from materials having two
martensitic energy wells (such as for orthorhombic to monoclinic
transformations), but without any restrictions on the possible microstructure,
showing how a generalization of the Hadamard jump condition can be applied at
the intergrain boundary to show that a pure phase in either grain is impossible
at minimum energy.Comment: ESOMAT 2015 Proceedings, to appea
Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams
In a cold-dark-matter universe, cosmological structure formation proceeds in
rough analogy to origami folding. Dark matter occupies a three-dimensional
'sheet' of free- fall observers, non-intersecting in six-dimensional
velocity-position phase space. At early times, the sheet was flat like an
origami sheet, i.e. velocities were essentially zero, but as time passes, the
sheet folds up to form cosmic structure. The present paper further illustrates
this analogy, and clarifies a Lagrangian definition of caustics and streams:
caustics are two-dimensional surfaces in this initial sheet along which it
folds, tessellating Lagrangian space into a set of three-dimensional regions,
i.e. streams. The main scientific result of the paper is that streams may be
colored by only two colors, with no two neighbouring streams (i.e. streams on
either side of a caustic surface) colored the same. The two colors correspond
to positive and negative parities of local Lagrangian volumes. This is a severe
restriction on the connectivity and therefore arrangement of streams in
Lagrangian space, since arbitrarily many colors can be necessary to color a
general arrangement of three-dimensional regions. This stream two-colorability
has consequences from graph theory, which we explain. Then, using N-body
simulations, we test how these caustics correspond in Lagrangian space to the
boundaries of haloes, filaments and walls. We also test how well outer caustics
correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS.
9 pages, 5 figure
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