43rd KPP Combinatorics Workshop - Dec 18 2010

secttitle: | Dec 18 (Saturday), 2010 |

Time: | 11am - 5:30pm ( Dinner from 6pm ) |

Location: | College of Natural Sciences, Building 1, Room 213 |

Schedule | ||

11:00 - 11:50 | Dan Drake KAIST, Daejeon | Paths, polynomials, partitions, and paths |

12:00 | Lunch | |

1:30 -2:20 | Byungchan Kim Seoul National University, Seoul | Combinatorics of Partial Theta Functions |

2:30 -3:20 | David Cariolaro Xi’an Jiaotong-Liverpool University, China | Excessive Factorizations |

3:40 - 4:30 | Jung Wook Lim Postech University, Pohang | Zero-divisor graphs over a commutative ring |

4:40 - 5:30 | Shinya Fujita Gunma National College of Technology, Japan | Rainbow Ramsey Numbers and Related Topics |

6:00 | Dinner |

## Abstracts

Dan Drake

Paths, polynomials, partitions, and paths

Paths, polynomials, partitions, and paths

We start with weighted Motzkin paths, then discuss how those paths are
the basis of a combinatorial theory of orthogonal polynomials. Four sets
of "classical" orthogonal polynomials naturally describe a hierarchy
of noncrossing complete matchings, complete matchings, set partitions,
and permutations. From matchings and set partitions, we move on to
consider$k-distant noncrossing matchings and set partitions, and end
with a bijection from 2-distant noncrossing matchings to weighted
Motzkin paths, thus returning us to where we started.

Byungchan Kim

Combinatorics of Partial Theta Functions

Combinatorics of Partial Theta Functions

A partial theta function is a sum of the form
. Combinatorially, identities
containing partial theta function are very interesting since they indicate
what remains after numerous cancellations of certain kinds of partitions. In
this talk, we will discuss the combinatorics of some identities involving
partial theta functions in Ramanujan's lost notebook.

David Cariolaro

Excessive Factorizations

Excessive Factorizations

An

*excessive factorization*of a graph is a minimum set of perfect matchings of whose union is . The*excessive index*of is the number of perfect matchings in an excessive factorization of (or if the graph does not admit an excessive factorization). More generally, let be a positive integer. An*excessive -factorization*of a graph is a minimum set of matchings, all of size exactly , whose union is the edge set of . The*excessive -index*of is the number of matchings in an excessive -factorization of (or if the graph does not admit an excessive -factorization).In this talk we shall aim to give an introduction to the theory of excessive (and excessive -)factorizations.

The content of this talk is joint work with Arrigo Bonisoli, Hung-Lin Fu, Giuseppe Mazzuoccolo and Romeo Rizzi.

Jung Wook Lim

Zero-divisor graphs over a commutative ring

Zero-divisor graphs over a commutative ring

In 1988 , Beck first
introduced the concept of zero-divisor graphs over a commutative ring with identity.
Especially, he was interested in coloring problems. Later, in 1999, Anderson and
Livingston defined a new concept of zero-divisor graphs over commutative rings.
Following Anderson et al., the

*zero-divisor graph*of a commutative ring , denoted by , is the graph with vertices , the set of nonzero zero-divisors of , and for distinct , the vertices and are adjacent if and only if . In the first part of this talk, I will survey some properties of zero-divisor graphs. Next, I will prove some basic results about the zero-divisor graphs of the rings and , where is an extension of commutative rings with the same total quotient ring .
Shinya Fujita

Rainbow Ramsey Numbers and Related Topics

Rainbow Ramsey Numbers and Related Topics

Gallai-colorings of complete graphs - edge colorings such that no triangle
is colored with three distinct colors - occur in various contexts such as the
theory of partially ordered sets (in Gallai's original paper), information theory and the theory of perfect graphs. A basic property of Gallai-colorings
with at least three colors is that at least one of the color classes must span
a disconnected graph. We are interested here in whether this or a similar
property remains true if we consider colorings that do not contain a rainbow
copy of a fixed graph F. In this talk, I will introduce some recent results on
this topic, and I will also mention a related topic concerning rainbow
ramsey numbers.