Mikio Kano

Discrete Geometry on Red and Blue Points in the Plane — A Survey —

In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider balanced subdivision problems, geometric graph problems, graph embedding problems, Gallai-type problems and others.

Factors and factorizations of graphs—a survey

A degree factor of a graph is either an r-factor (regular of degree r) or an [m, n]-factor (with each degree between m and n). In a component factor, each component is a prescribed graph. Both kinds of factors are surveyed, and also corresponding factorizations.

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge-colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. We have to point out that there are a lot of results on Ramsey type problem of monochromatic and heterochromatic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not include them, either.

Some results on odd factors of graphs

A {1, 3, …,2n − 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, …, 2n − 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, …, 2n − 1}-factor.

[a,b]-factorization of a graph

This is the authors version of a work that was accepted for publication in Journal of Graph Theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Graph Theory, VOLUME 9, ISSUE 1, 129-146, 1985, 10.1002/jgt.3190090111.

Balanced partitions of two sets of points in the plane

Abstract We prove the following theorem. Let m≥2 and q≥1 be integers and let S and T be two disjoint sets of points in the plane such that no three points of S∪T are on the same line, |S|=2q and |T|=mq. Then S∪T can be partitioned into q disjoint subsets P1,P2,…,Pq satisfying the following two conditions: (i) conv(Pi)∩conv(Pj)=φ for all 1≤i

Straight line embeddings of rooted star forests in the plane

Abstract For every 1⩽i⩽n, let Ti be a rooted star with root vi, where vi is not necessarily its center. Then the union F=T1∪T2∪⋯∪Tn is called a rooted star forest with roots v1,v2,…,vn. Let P be a set of |F| points in the plane in general position containing n specified points p1,p2,…,pn, where |F| denotes the order of F. Then we show that there exists a bijection φ:V(F)→P such that φ(vi)=pi for all 1⩽i⩽n, φ(x) and φ(y) are joined by a straight-line segment if and only if x and y are joined by an edge of F, and such that no two straight-line segments intersect except at their common end-point.

Encompassing colored planar straight line graphs

Consider a planar straight line graph (PSLG), G, with k connected components, k>=2. We show that if no component is a singleton, we can always find a vertex in one component that sees an entire edge in another component. This implies that when the vertices of G are colored, so that adjacent vertices have different colors, then (1) we can augment G with k-1 edges so that we get a color conforming connected PSLG; (2) if each component of G is 2-edge connected, then we can augment G with 2k-2 edges so that we get a 2-edge connected PSLG. Furthermore, we can determine a set of augmenting edges in O(nlogn) time. An important special case of this result is that any red-blue planar matching can be completed into a crossing-free red-blue spanning tree in O(nlogn) time.

Ranking the Vertices of a Paired Comparison Digraph

A paired comparison digraph

SEMI-BALANCED PARTITIONS OF TWO SETS OF POINTS AND EMBEDDINGS OF ROOTED FORESTS

D = ( V,A )