Jin Akiyama

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.

Simple alternating path problem

Abstract Let A be a set of 2 n points in general position on a plane, and suppose n of the points are coloured red while the remaining are coloured blue. An alternating path P of A is a sequence p 1 , p 2 ,…, p 2 n of points of A such that p 2 i is blue and p 2 i +1 is red. P is simple if it does not intersect itself. We determine the condition under which there exists a simple alternating path P of A for the case when the 2 n points are the vertices of a convex polygon. As a consequence an O( n 2 ) algorithm to find such an alternating path (if it exists) is obtained.

Balancing signed graphs

Abstract A signed graph based on F is an ordinary graph F with each edge marked as positive or negative. Such a graph is called balanced if each of its cycles includes an even number of negative edges. Psychologists are sometimes interested in the smallest number d=d(G) such that a signed graph G may be converted into a balanced graph by changing the signs of d edges. We investigate the number D(F) defined as the largest d(G) such that G is a signed graph based on F. We prove that 1 2 m− nm ≤D(F)≤ 1 2 m for every graph F with n vertices and m edges. If F is the complete bipartite graph with t vertices in each part, then D(F)≤ 1 2 t 2 −ct 3 2 for some positive constant c.

Dudeney Dissection of Polygons

Given an equilateral triangle A and A Square B of the same area, Henry E. Dudeney introduced A partition of A into parts that tan be reassembled in some way, without turning over the surfaces, to form B. An examination of Dudeney’s method of partition motivates us to introduce the notion of Dudeney dissections of various polygons to other polygons.

Combinatorial Geometry and Graph Theory

This volume consists of the refereed papers presented at the Indonesia-Japan Joint Conference on Combinatorial Geometry and Graph Theory (IJCCGGT 2003), held on September 13 16, 2003 at ITB, Bandung, Indonesia. This conf- ence can also be considered as a series of the Japan Conference on Discrete and Computational Geometry (JCDCG), which has been held annually since 1997. The ?rst ?ve conferences of the series were held in Tokyo, Japan, the sixth in Manila, the Philippines, in 2001, and the seventh in Tokyo, Japan in 2002. The proceedings of JCDCG 1998, JCDCG 2000 and JCDCG 2002 were p- lished by Springer as part of the series Lecture Notes in Computer Science: LNCS volumes 1763, 2098 and 2866, respectively. The proceedings of JCDCG 2001 were also published by Springer as a special issue of the journal Graphs and Combinatorics, Vol. 18, No. 4, 2002. TheorganizersaregratefultotheDepartmentofMathematics,InstitutTek- logi Bandung (ITB) and Tokai University for sponsoring the conference. We also thank all program committee members and referees for their excellent work. Our big thanks to the principal speakers: Hajo Broersma, Mikio Kano, Janos Pach andJorgeUrrutia.Finally,ourthanksalsogoestoallourcolleagueswhoworked hard to make the conference enjoyable and successful. August 2004 Jin Akiyama Edy Tri Baskoro Mikio Kano Organization The Indonesia-Japan Joint Conference on Combinatorial Geometry and Graph Theory (IJCCGGT) 2003 was organized by the Department of Mathematics, InstitutTeknologiBandung(ITB)IndonesiaandRIED,TokaiUniversity,Japan

Radial Perfect Partitions of Convex Sets in the Plane

In this paper we study the following problem: how to divide a cake among the children attending a birthday party such that all the children get the same amount of cake and the same amount of icing. This leads us to the study of the following. A perfect k-partitioning of a convex set S is a partitioning of S into k convex pieces such that each piece has the same area and \(\frac{1}{k}\) of the perimeter of S . We show that for any k, any convex set admits a perfect k-partitioning. Perfect partitionings with additional constraints are also studied.

Some combinatorial problems

Abstract There are many interesting and sophisticated problems posed in the IMO, Putnam and domestic Olympiads. Some of these problems have deep mathematical background, nice generalizations, and lead to new areas of research in combinatorics. We investigate several topics in this category and mention some results and open problems.

Path chromatic numbers of graphs

Let the finite, simple, undirected graph G = (V(G), E(G)) be vertex-colored. Denote the distinct colors by 1,2,…,c. Let Vi be the set of all vertices colored j and let <Vi be the subgraph of G induced by Vi. The k-path chromatic number of G, denoted by χ(G; Pk), is the least number c of distinct colors with which V(G) can be colored such that each connected component of Vi is a path of order at most k, 1 ⩽ i ⩽ c. We obtain upper bounds for χ(G; Pk) and χ(G; P∞) for regular, planar, and outerplanar graphs.

Almost‐regular factorization of graphs

For integers a and b, 0 ⩽ a ⩽ a ⩽ b, an [a, b]-graph G satisties a ⩽ deg(x, G) ⩽ b for every vertex x of G, and an [a, b]-factor is a spanning subgraph F such that a ⩽ deg(x, F) ⩽ b for every vertex x of F. An [a, b]-factor is almost-regular if b = a + 1. A graph is [a, b]-factorable if its edges can be decomposed into [a, b]-factors. When both K and t are positive integers and s is a nonnegative integer, we prove that every [(12K + 2)t + 2ks, (12k + 4)t + 2ks]-graph is [2k,2k + 1]-factorable. As its corollary, we prove that every [r.r + 1]-graph with r ⩾ 12k2 + 2k is [2k + 1]-factorable, which is a partial extension of the two results, one by Thomassen and the other by Era.

On the size of graphs with complete-factors

A spanning subgraph H of a graph G is called a Kl-factor if each component of H is isomorphic to the complete graph of order l. We determine the minimum size for any graph to have a Kl-factor. Relating this result, we give a new short proof of the Erdos-Gallai theorem on the maximum size of graphs with at most β independent edges.