Mariko Hagita

Toughness and the existence of k -factors. IV

In a paper with the same title [3], we proved Chvatals conjecture thatk-tough graphs havek-factors if they satisfy trivial necessary conditions. In this paper, we prove the following stronger result: Suppose|V(G)| ≥ k + 1,k ⋅ |V(G)| even, and|S| ≥ k ⋅ w(G − S) − 7/8k ifw(G − S) ≥ 2, wherew(G − S) is the number of connected components ofG − S. ThenG has ak-factor.

Covering vertices of a graph by k disjoint cycles

Abstract Let d, k and n be three integers with k⩾3, d⩾4k−1 and n ⩾3 k . We show that if d ( x )+ d ( y )⩾ d for each pair of nonadjacent vertices x and y of a graph G of order n , then G contains k vertex-disjoint cycles converting at least min{ d , n } vertices of G .

CryptMT3 Stream Cipher

CryptMT version 3 (CryptMT3) is a stream cipher obtained by combining a large LFSR and a nonlinear filter with memory using integer multiplication. Its period is proved to be no less than 219937? 1, and the 8-bit output sequence is at least 1241-dimensionally equidistributed. It is one of the fastest stream ciphers on a CPU with SIMD operations, such as Intel Core 2 Duo.

A fast stream cipher with huge state space and quasigroup filter for software

Recent personal computers have high-spec CPUs and plenty of memory. The motivation of this study is to take these advantages in designing a tough and fast key-stream generator. Natural controversies on using a large state space for a generator are (1) effectiveness is unclear, (2) slower generation speed, (3) expensive initialization, and (4) costs in a hardware implementation. Our proposal is to combine a linear feedback shift register (LFSR) and a uniform quasigroup filter with memory of wordsize. We prove theorems which assure the period and the distribution property of such generators, answering to (1). As for (2), the generation speed of a LFSR is independent of the state size. In addition, we propose a filter based on integer multiplication, which is rather fast in modern CPUs. We analyze the algebraic degree of such filters. We answer to (3) by a simple trick to use another small generator to initialize LFSR while outputting. We have no answer to (4), but comment that recent hardwares tend to have larger memory and sophisticated instructions. As a concrete example, we propose CryptMT stream generator with period (no less than) 219937 - 1, 1241-dimensional equidistribution property, which is sometimes faster than SNOW2.0 in modern CPUs.

The Diameters of Some Transition Graphs Constructed from Hamilton Cycles

Abstract. A complete undirected graph of order n has Hamilton cycles. We consider the diameter of a transition graph whose vertices correspond to those Hamilton cycles and any of two vertices are adjacent if and only if the corresponding Hamilton cycles can be transformed each other by exchanging two edges. Moreover, we consider several transition graphs related to it.

Bijections Between Group Rings Preserving Character Sums

Generalizing an idea in [13], we exhibit some, in general nonhomomorphic, bijections between finite groups which preserve the absolute value of character sums. As a consequence, the existence of a single difference set, relative difference set, building set etc. in certain groups implies the existence of such objects in many other groups.

A Note on Difference Sets

LetDbe a (v,k,?)-difference set in a groupG. Assume thatGhas a normal subgroupNsuch thatG/Nis cyclic or nearly cyclic. Under the self-conjugacy assumption on exp(G/N), we shall give bounds on |N| and?. The theorem is applicable to a wider variety of parameters for groups, not necessarily abelian. These bounds exclude a (96, 20, 4)-difference set inZ/4Z×Z/8Z×Z/3ZorZ/2Z×Z/2Z×Z/8Z×Z/3Z, which were recently proved by Arasuet al. [1996, K. T. Arasu, J. A. Davis, J. Jedwab, S. L. Ma, and R. L. McFarland,in“Exponent Bounds for a Family of Abelian Difference Sets” (K. T. Arasu, J. F. Dillon, K. Harada, S. K. Sehgal, and R. L. Solomon, Eds.), pp. 129?143. DeGruyter Verlag, Berlin/New York].

Cycles having the same modularity and removable edges in 2-connected graphs

In this paper, we consider 2-connected multigraphs in which every cycle has length congruent to a modulo b (b>=2). We prove that there exists such a multigraph which is homomorphic to a graph with minimum degree at least three only if a=0, and that there exists such a graph only if a=0 and b=2. We also study the distribution of paths whose internal vertices have degree exactly two, and show a relation between these paths and edges in a 2-connected graph whose deletion results in a 2-connected graph.

Foldings of Difference Sets in Abelian Groups

Abstract. In this paper, we show that under some conditions the existence of a difference set in G implies the existence of another difference set with the same parameters in G′, where G and G′ are abelian groups of the same order. This explains why there are more difference sets in abelian groups of low exponent and high rank than in those of high exponent and low rank.

A Technique for Ranking and Visualization of Crowd-Powered Subjective Evaluations

We previously presented a crowd-powered digital contents evaluation system. This system shows a lot of pictures to the answerers and ask them to input the evaluations. It preferentially selects pictures which are predicted to be highly or poorly evaluated to the answerers, based on our assumption that high or poor evaluations are more informative results comparing with moderate evaluations. We have applied an interactive genetic algorithm in our system to select such pictures. This paper presents a technique for ranking and visualization for the evaluation results collected by our system. The presented technique calculates scores of all contents and uses for the ranking. Here, it may happen that some pictures are shown to no answerers while using our evaluation system. Our technique presented in this paper estimates the evaluation of such pictures shown to no answerers, and finally complete the ranking of all the pictures. The paper also presents the visualization tool for the ranking of pictures, and our experiment to demonstrate the effectiveness of our technique.