Shinya Fujita

Rainbow Generalizations of Ramsey Theory: A Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

A pair of forbidden subgraphs and perfect matchings

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to be H-free if no graph in H is an induced subgraph of G. We completely characterize the set H such that every connected H-free graph of sufficiently large even order has a perfect matching in the following cases.(1) Every graph in H is triangle-free. (2) H consists of two graphs (i.e. a pair of forbidden subgraphs).A matching M in a graph of odd order is said to be a near-perfect matching if every vertex of G but one is incident with an edge of M. We also characterize H such that every H-free graph of sufficiently large odd order has a near-perfect matching in the above cases.

Recent Results on Disjoint Cycles in Graphs

Abstract Recent progresses on degree conditions and vertex-disjoint cycles in graphs will be reviewed.

The Balanced Decomposition Number and Vertex Connectivity

The balanced decomposition number

Connectivity keeping edges in graphs with large minimum degree

f(G)

A pair of forbidden subgraphs and perfect matchings in graphs of high connectivity

of a graph

Non-separating even cycles in highly connected graphs

G

Minimally contraction-critically 6-connected graphs

was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339-3344]. A balanced coloring of a graph

Constructing connected bicritical graphs with edge-connectivity 2

G

Disjoint Even Cycles Packing

is a coloring of some of the vertices of