The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph is being studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is no smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.Comment: 6 pages, 3 figure

Computerized pattern making focus on fitting to 3D human body shapes

Purpose - This paper aims to describe the development of a method of constructing three-dimensional (3D) human body shapes that include a degree of ease for purpose of computerized pattern making. Design/methodology/approach - The body shape could be made with ease allowance to an individual's unique body shape using sweep method and a convex method. And then generates tight skirt patterns for the reconstructed virtual body shape using a computerized pattern making system. Findings - This paper obtains individual patterns using individually reconstructed 3D body shapes by computerized pattern development. In these patterns, complex curved lines such as waist lines and dart lines are created automatically using the developed method. The method is successfully used to make variations of a tight skirt to fit different size women. The author also used the method to make other skirts of various designs. Originality/value - The method described in this paper is useful for making patterns and then garments, without the need for the garments to be later adjusted for the subject.ArticleInternational Journal of Clothing Science and Technology. 22(1):16-24 (2010)journal articl

The competition hypergraphs of doubly partial orders

Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting results have been obtained. Continuing in the same spirit, we study the competition hypergraph, an interesting variant of the competition graph, of a doubly partial order. Though it turns out that the competition hypergraph of a doubly partial order is not always interval, we completely characterize the competition hypergraphs of doubly partial orders which are interval.Comment: 12 pages, 6 figure

Universal Seesaw Mass Matrix Model with an S_3 Symmetry

Stimulated by the phenomenological success of the universal seesaw mass matrix model, where the mass terms for quarks and leptons f_i (i=1,2,3) and hypothetical super-heavy fermions F_i are given by \bar{f}_L m_L F_R +\bar{F}_L m_R f_R + \bar{F}_L M_F F_R + h.c. and the form of M_F is democratic on the bases on which m_L and m_R are diagonal, the following model is discussed: The mass terms M_F are invariant under the permutation symmetry S_3, and the mass terms m_L and m_R are generated by breaking the S_3 symmetry spontaneously. The model leads to an interesting relation for the charged lepton masses.Comment: 8 pages + 1 table, latex, no figures, references adde

Evolution of the Yukawa coupling constants and seesaw operators in the universal seesaw model

The general features of the evolution of the Yukawa coupling constants and seesaw operators in the universal seesaw model with det M_F=0 are investigated. Especially, it is checked whether the model causes bursts of Yukawa coupling constants, because in the model not only the magnitude of the Yukawa coupling constant (Y_L^u)_{33} in the up-quark sector but also that of (Y_L^d)_{33} in the down-quark sector is of the order of one, i.e., (Y_L^u)_{33} \sim (Y_L^d)_{33} \sim 1. The requirement that the model should be calculable perturbatively puts some constraints on the values of the intermediate mass scales and tan\beta (in the SUSY model).Comment: 21 pages, RevTex, 10 figure

A Unified Description of Quark and Lepton Mass Matrices in a Universal Seesaw Model

In the democratic universal seesaw model, the mass matrices are given by \bar{f}_L m_L F_R + \bar{F}_L m_R f_R + \bar{F}_L M_F F_R (f: quarks and leptons; F: hypothetical heavy fermions), m_L and m_R are universal for up- and down-fermions, and M_F has a structure ({\bf 1}+ b_f X) (b_f is a flavour-dependent parameter, and X is a democratic matrix). The model can successfully explain the quark masses and CKM mixing parameters in terms of the charged lepton masses by adjusting only one parameter, b_f. However, so far, the model has not been able to give the observed bimaximal mixing for the neutrino sector. In the present paper, we consider that M_F in the quark sectors are still "fully" democratic, while M_F in the lepton sectors are partially democratic. Then, the revised model can reasonably give a nearly bimaximal mixing without spoiling the previous success in the quark sectors.Comment: 7 pages, no figur

S_3 Symmetry and Neutrino Masses and Mixings

Based on a universal seesaw mass matrix model with three scalars \phi_i, and by assuming an S_3 flavor symmetry for the Yukawa interactions, the lepton masses and mixings are investigated systematically. In order to understand the observed neutrino mixing, the charged leptons (e, \mu, \tau) are regarded as the 3 elements (e_1, e_2, e_3) of S_3, while the neutrino mass-eigenstates are regarded as the irreducible representation (\nu_\eta, \nu_\sigma, \nu_\pi) of S_3, where (\nu_\pi, \nu_\eta) and \nu_\sigma are a doublet and a singlet, respectively, which are composed of the 3 elements (\nu_1, \nu_2, \nu_3) of S_3.Comment: 16 pages, no figure, version to appear in EPJ-