A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

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Journal of Combinatorial Theory, Series B 100 (2010) 161-170. doi:10.1016/j.jctb.2009.05.006Received by publisher: 2008-03-17Harvest Date: 2016-01-04 12:21:10DOI: 10.1016/j.jctb.2009.05.006Page Range: 161-17

The main theorem of this paper establishes conditions under which the "chaos game" algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously. General iterated function systemsThroughout this paper (X, d X ) is a complete metric space.Definition 1 If f m : X → X, m = 1, 2, . . . , M, are continuous mappings, then F = (X; f 1 , f 2 , ..., f M ) is called an iterated function system (IFS).By slight abuse of terminology we use the same symbol F for the IFS, the set of functions in the IFS, and for the following mapping. Letting 2 X

An effective method is given for computing the Hausdorff dimension of the boundary of a self‐similar digit tile T in n‐dimensional Euclidean space: dimH(∂T)=logλlogc where 1/c is the contraction factor and λ is the largest eigenvalue of a certain contact matrix first defined by Gröchenig and Haas.

A theory of replicating tessellation of R is developed that simultaneously generalizes radix representation of integers and hexagonal addressing in computer science. The tiling aggregates tesselate Euclidean space so that the (m + 1)st aggregate is, in turn, tiled by translates of the ruth aggregate, for each m in exactly the same way. This induces a discrete hierarchical addressing systsem on R'. Necessary and sufficient conditions for the existence of replicating tessellations are given, and an efficient algorithm is provided to determine whether or not a replicating tessellation is induced. It is shown that the generalized balanced ternary is replicating in all dimensions. Each replicating tessellation yields an associated self-replicating tiling with the following properties: (1) a single tile T tesselates R periodically and (2) there is a linear map A, such that A(T) is tiled by translates of T. The boundary of T is often a fractal curve.Recall that a lattice A determines a tessellation by polytopal Voronoi cells where the Voronoi cell of the lattice point z is defined by {y ' Ig z[ < lY zl for all z A}. Let V, denote the union of the Voronoi cells corresponding to the lattice points ofThe definition of rep-tiling is equivalent to (1) V, tiles by translation by the sublattice A '+1 (A), for each m > 0, and (2) every point of l' lies in V, for some m. The set S, (or the corresponding V,) is called the m-aggregate of the pair (A, S). If S induces a replicating tessellation, then the (m + 1)-aggregate is tiled by IS[ copies of the maggregate for each m > 0. More precisely, So S and S,,+1 is the disjoint union zA,+(S) for all m > 0. Hence, we have the term "replicating." edu

Several problems concerning the distribution of cycle lengths in a graph have been proposed by P. Erdös and colleagues. In this note two variations of the following such question are answered. In a simple graph where every vertex has degree at least three, must there exist two cycles whose lengths differ by one or two?