Ge 2k Research Articles

Circular chromatic index of Cartesian products of graphs

AbstractThe circular chromatic index of a graph G, written $\chi_{c}'(G)$, is the minimum r permitting a function $f : E(G)\rightarrow [0,r)$ such that $1 \le | f(e)-f(e')|\le r - 1$ whenever e and $e'$ are incident. Let $G = H$ □ $C_{2m +1}$, where □ denotes Cartesian product and H is an $(s-2)$‐regular graph of odd order, with $s \equiv 0 \, {\rm mod}\, 4$ (thus, G is s‐regular). We prove that $\chi_{c}'(G) \ge s +\lfloor \lambda(1 - 1/s)\rfloor^{-1}$, where $\lambda$ is the minimum, over all bases of the cycle space of H, of the maximum length of a cycle in the basis. When $H = C_{2k +1}$ and m is large, the lower bound is sharp. In particular, if $m \ge 3k + 1$, then $\chi_{c}'(C_{2k +1}$ □ $C_{2m + 1})=4 + \lceil {3k/2} \rceil^{-1}$, independent of m. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 7–18, 2008

RSK Insertion for Set Partitions and Diagram Algebras

We give combinatorial proofs of two identities from the representation theory of the partition algebra ${\Bbb C} A_k(n)$, $n \ge 2k$. The first is $n^k = \sum_\lambda f^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$, $f^\lambda$ is the number of standard tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of "vacillating tableaux" of shape $\lambda$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_\lambda (m_k^\lambda)^2$, where $B(2k)$ is the number of set partitions of $\{1, \ldots, 2k\}$. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.

Maximal pattern complexity for discrete systems

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dotsc over a finite alphabet, the authors introduced a new notion of complexity called maximal pattern complexity defined by p_\alpha^*(k):=\sup_\tau\sharp\{\alpha_{n+\tau(0)}\alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)};n=0,1,2,\dotsc\} where the supremum is taken over all sequences of integers 0=\tau(0)<\tau(1)<\dotsc<\tau(k-1) of length k. The authors proved that \alpha is aperiodic if and only if p_\alpha^*(k)\ge 2k for every k=1,2,\dotsc. A word \alpha with p_\alpha^*(k)=2k for every k\geq 1 is called pattern Sturmian. In this paper, we give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations. We also investigate the maximal pattern complexity of various discrete dynamical systems including irrational rotations on the circle, and self-similar systems generated by substitutions. We show that, for each irrational rotation on the circle, there exists a twofold partition of the circle, with respect to which the system generated has full maximal pattern complexity with probability one. Using the arithmetic properties of the underlying numeration system associated to a substitution dynamical system, we prove that the maximal pattern complexity of the fixed point of the Rauzy substitution 1\mapsto 12, 2\mapsto 13, 3\mapsto 1 has exponential growth. It is well known that the system generated by the Rauzy substitution is isomorphic in measure to an irrational rotation on the 2-torus.

A degree estimate for subdivision surfaces of higher regularity

Subdivision algorithms can be used to construct smooth surfaces from control meshes of arbitrary topological structure. In contrast to tangent plane continuity, which is well understood, very little is known about the generation of subdivision surfaces of higher regularity. This work presents a degree estimate for piecewise polynomial subdivision surfaces saying that curvature continuity is possible only if the bi-degree d d of the patches satisfies d ≥ 2 k + 2 d \ge 2k+2 , where k k is the order of smoothness on the regular part of the surface. This result applies to any stationary or non-stationary scheme consisting of masks of arbitrary size provided that some generic symmetry and regularity assumptions are fulfilled.