**MME Staff and Graduate Student Colloquium 2018**

Date: Friday 30 November 2018

Time: 4.30 pm – 8.30 pm

Venue: TR201 (Math Edn) & TR203 (Math)

Registration closes : 15 November 2018

## Mathematics Abstracts of Presentations

*On the skewness of Cartesian products with trees *

Tay Eng Guan

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. In this talk, we show that skewness is additive for the Zip product under certain conditions. We then present results on lower bounds and exact values of Cartesian products of graphs with trees. This is work done by Ouyang Zhangdong, Dong Fengming and Tay Eng Guan during the former’s visit to NIE in 2017/18.

*On the existence of real roots of chromatic polynomials of hypergraphs*

Zhang Ruixue

A mixed hypergraph $\mathcal{H}$ is a triple $(\mathcal{V}, \mathcal{C}, \mathcal{D})$, where $\mathcal{V}$ is a finite set, $\mathcal{C}$ and $\mathcal{D}$ are the subsets of $\{e\subseteq \mathcal{V}: |e|\geq 1\}$. For any positive integer $\lambda$, a proper $\lambda$-coloring of $\mathcal{H}$ is a mapping $\phi: \mathcal{V}\rightarrow \{1, 2, \cdots, \lambda\}$ such that $1\leq |\{\phi(v): v\in e\}|\leq |e|-1$ for each $e\in \mathcal{C}$ and $|\{\phi(v): v\in e\}|\geq 2$ for each $e\in \mathcal{D}$. Let $P_{mix}(\mathcal{H}, \lambda)$ denote the polynomial that counts the number of proper $\lambda$-colorings of $\mathcal{H}$ whenever $\lambda$ is a positive integer. In this talk, we provide some sufficient conditions on a mixed hypergraph $\mathcal{H}$ such that $P_{mix}(\mathcal{H}, \lambda)$ has no real root in the intervals $(-\infty, 0)$ and $(0, 1)$. This result is an extension of known results on the zero-free intervals of chromatic polynomials of hypergraphs and graphs.

*An Open Problem in Domain Theory*

Ng Kok Min

Every RB-domain is an FS-domain. However, whether the converse is true is an open problem in Domain Theory. Lawson (2008) showed that the pointed domain of closed balls, ${\bf B}(\mbox{R}^n)$, of the familiar Euclidean space $(\mbox{R}^n)$ is an FS-domain. While much have been known of the Euclidean space, it is surprising that it remains unknown if ${\bf B}(\mbox{R}^n)$ is an RB-domain for $n$ greater than 1. This then naturally leads to the suspicion that ${\bf B}(\mbox{R}^n)$ can serve as the counterexample to the open problem. In this talk, we discuss the problem and some of its related progress.

*On the Evenly Spaced Topology on Zs*

Jeremy Ibrahim Bin Abdul Gafar

In number theory, the infinitude of primes is a well-known result that was first established and recorded by Euclid around 300 B.C. By all standards of rigour in modern mathematics, Euclid's style of reasoning still stands as an excellent model for the proof method by contradiction. Another less known but nonetheless curiously interesting proof was due to Hillel Furstenberg. Hillel Furstenberg’s proof of the infinitude of primes is a celebrated topological proof that the set of integers contains infinitely many prime numbers. The proof was published in 1955 in the American Mathematical Monthly while Furtsenberg was still an undergraduate student at Yeshiva University. Central in Furstenberg's proof is a type of topology defined on the setof natural numbers which is called evenly spaced topologies on Z. Later, S. Golomb continued along this line and look at another evenly spaced topology defined on Z and its properties in connection with its number-theoretic implications expounded.

In my Final Year Project, I studied Furstenberg's proof of the infinitude of primes, noting which properties of the evenly spaced topology are important in the proof. I also carried out a further study into the topological properties of the Furstenberg's evenly spaced topology, other than those involved in the proof.

*Hamilton Path Decompositions of Complete Multipartite Graphs*

Hang Hao Chuien

There has been interest in problems concerning decomposition of graphs into Hamilton cycles and into Hamilton paths, for many years. In 1976, Laskar and Auerbach showed that a complete multipartite graph can be decomposed into Hamilton cycles if and only if it is regular of even degree. We prove a corresponding result for decompositions of complete multipartite graphs into Hamilton paths. In particular, we prove that a complete multipartite graph $K$ with $n>1$ vertices and $m$ edges can be decomposed into edge-disjoint Hamilton paths if and only if $m/(n-1)$ is an integer and the maximum degree of $K$ is at most $2m/(n-1)$. This is joint work with Darryn Bryant and Sara Herke.