# Registered Data

## [00575] Factors and Cycles

**Session Date & Time**: 5B (Aug.25, 10:40-12:20)**Type**: Proposal of Minisymposium**Abstract**: Factors and cycles are well established subjects in the field of graph theory. They are basic and fundamental problems: for a given graph $G$, taking a regular graph as a spanning subgraph of $G$. On the other hand, factors and cycles have several applications in contexts including error correction coding theory, scheduling problems, wireless networking, and many others. Regardless of efforts by mathematicians, there are many problems on factors and cycles which are not solved yet. This minisymposium intends to bring pioneer researchers to present their very recent discoveries on factors, cycles and related topics.**Organizer(s)**: Shoichi Tsuchiya**Classification**:__05C38__,__05C45__,__05C70__**Speakers Info**:- Jun Fujisawa (Keio University)
- Suil O (The State University of New York, Korea)
- Takamasa Yashima (Seikei university)
- Masahiro Sanka (Keio university)

**Talks in Minisymposium**:**[02216] Recent progress on distance matching extension in graphs on surfaces****Author(s)**:**Jun Fujisawa**(Keio University)

**Abstract**: A graph $G$ is said to be distance $d$ matchable if, for any matching $M$ of $G$ in which edges are pairwise at least distance $d$ apart, there exists a perfect matching of $G$ which contains $M$. If we choose a class of graphs $A$ and an integer $d$ appropriately, then we may observe a phenomenon that every graph in $A$ is distance $d$ matchable. In this talk recent results concerning this phenomenon is introduced.

**[02275] Toughness and forbidden subgraphs for graphs to be hamiltonian****Author(s)**:**Masahiro Sanka**(Keio University)

**Abstract**: We say that a graph $G$ is $t$-tough if for each vertex cut $S \subset V(G)$, the number of components of $G-S$ does not exceed $\frac{|S|}{t}$. In 1973, Chvátal has conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph is hamiltonian. In this talk, we discuss some results on Chvátal's conjecture in $R$-free graphs for some graph $R$. In particular, we will present hamiltonicity of $2K_2$-free graphs and $(K_2 \cup kK_1)$-free graphs.

**[02338] On cycles and factors in graphs with large degree sum****Author(s)**:**Takamasa Yashima**(Seikei University)

**Abstract**: Numerous researchers have conducted investigations into a Hamiltonian cycle and related objects in both general graphs and bipartite graphs. In this talk, we will focus on two specific topics related to this field. The first topic concerns the problem of finding a spanning subgraph, also known as a factor, that satisfies certain degree constraints. The second topic concerns the problem of finding a Hamiltonian cycle containing a pre-specified set of independent edges, or a matching.

**[02696] Eigenvalues and factors in regular graphs****Author(s)**:**Suil Oh**(SUNY Korea)

**Abstract**: In this talk, we investigate spectral conditions for an (r-regular) graph G to guarantee the existence of a certain factor.