The cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the cyclic coloring complex of three different types of $k$-uniform hypergraphs. For the case of a complete $k$-uniform hypergraph, we show that the dimension of the $(n-k-1)^{st}$ homology group is given by a binomial coefficient. Further, we discuss a complex whose $r$-faces consist of all ordered set partitions $[B_1, ..., B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete $k$-uniform hypergraph $H$ and where $1 \in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of \C[x_1,...,x_n]/ \{x_{i_1}...x_{i_k} \mid i_{1}...i_{k} is a hyperedge of $H \}$. For the other two types of hypergraphs, star hypergraphs and diagonal hypergraphs, we show that the dimensions of the homology groups of their cyclic coloring complexes are given by binomial coefficients as well

EFFECT ON MILK PRODUCTION AND INCOME OVER FEED COST FROM FOLLOWING LESS THAN OPTIMUM MANAGEMENT STRATEGIES RELATED TO DAIRY COW REPLACEMENT

Livestock Production/Industries,

Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents

Asymmetric 22-colorings of graphs

We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors in such a way that the graph has no color preserving automorphisms. Also, we characterize all graphs which have the property that their edges can be $2$-colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface which induces a non-trivial color preserving automorphism of the graph

A Finite Element Method for the Fractional Sturm-Liouville Problem

In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $\alpha\in(1,2)$ on the unit interval $(0,1)$. It is based on novel variational formulations of the eigenvalue problem. Error estimates are provided for the finite element approximations of the eigenvalues. Numerical results are presented to illustrate the efficiency and accuracy of the method. The results indicate that the method can achieve a second-order convergence for both fractional derivatives, and can provide accurate approximations to multiple eigenvalues simultaneously.Comment: 30 pages, 7 figure