2019年度秋季総合分科会(於:金沢大学)

SUMMARY: This talk is concerned with approximation theory of analytic functions with prescribed decay on a strip region including the real axis. Such functions appear when we use numerical methods with variable transformations. Typical examples of such methods are provided by double-exponential (DE) sinc formulas for function approximation and DE formulas for numerical integration. They are based on variable transformations yielding double-exponential decay of functions on the real axis, which improves the accuracy of the approximation of functions or their integrals. It has been known that the formulas are nearly optimal on Hardy spaces with a double-exponential weight on the strip region, which are regarded as spaces of transformed functions by the variable transformations. However, optimal formulas have not been known explicitly so far. Then, we consider approximation theory of analytic functions in weighted Hardy spaces for a general weight on the strip region. More precisely, our objectives are (i) precise estimate of the best approximation of functions and their integrals in the space and (ii) finding general procedure to construct accurate approximation formulas based on the estimate. In particular, the first objective includes estimating an analogue of the linear n-width of the unit ball in the weighted Hardy space. We show some results about the first objective. Furthermore, with a view to the second objective, we have recently proposed a simple method for obtaining sampling points for approximating functions and numerical integration. This method is based on a convex minimization problem of a discrete energy. By solving the problem with a standard optimization technique, we obtain sampling points that realize accurate formulas for approximating functions and numerical integration. We also provide theoretical convergence analyses of the formulas via a duality theorem of the continuous counterpart of the minimization problem.

SUMMARY: In this talk, I will treat propagation phenomena in a free boundary problem of reaction diffusion equation of the form : \(u_t=u_{xx}+f(u)\) (\(t>0\), \(0<x<h(t)\)) where \(h(t)\) is the unknown moving boundary (free boundary) which is determined by the Stefan condition of the form \(h'(t)=-\mu u_x(t, h(t))\). In the work of Du and Lin (2010), this type of problem was introduced as a model which describes the spreading of new or invasive species. From this work, propagating phenomena in the free boundary problems attract more and more attention of mathematicians. Among the various studies on the free boundary problems, recent work of Du, Matsuzawa and Zhou (2014) showed that the profile of any spreading solution (which corresponds to the success of invasion) approaches a traveling wave solution associated with the free boundary problem, which was introduced by Du and Lou (2015) and is called semi-wave.

In this talk, I will give some recent study on propagation profiles of solutions for the free boundary problem of reaction-diffusion equation with some class of multi-stable nonlinearity. In particular, under certain condition, the profile of the spreading solution approaches a so-called propagating terrace. I will also present a result about radially symmetric solutions in high space dimensions.

This talk is based on some joint works with Dr. Yuki Kaneko (Japan Women’s University) and Professor Yoshio Yamada (Waseda University).

SUMMARY: A combinatorial game is a two-player game with perfect information (i.e., no hidden information), no chance moves (i.e., no probabilistic matter) and outcome restricted to (lose, win) and (draw, draw) for the two players who move alternately. Combinatorial games include well-known games such as Chess, Checkers, Go, and also include an elementary puzzles and games (enjoyed by children) such as Sudoku, tic-tac-toe and Geography. So far, a lot of combinatorial games are introduced and analyzed, and then lots of them are also considered on graphs as one of generalizations of those games. On the other hand, many mathematicians created combinatorial games on graphs which are deeply related to typical graph invariants. Thus, in this talk, we first briefly classify combinatorial games on graphs and introduce typical combinatorial games on graphs. Moreover, we also provide open problems on those games.

SUMMARY: A computation consists of algorithm, which is a sequence of basic operations. When you consider an algorithm, you assume some computation model that has “usual” arithmetic operations. On the other hand, when you enjoy a puzzle, you have to find an algorithm by combining reasonable basic operations to its goal. From the viewpoint of theoretical computer science, puzzles give us some insight to computation and computational complexity classes in various way.

Some puzzles and games give reasonable characterizations to computational complexity classes. For example, “pebble game” is a classic model that gives some complexity classes in a natural way, and “constraint logic” is recent model that succeeds to solve a long standing open problem due to Martin Gardner that asks the computational complexity of sliding block puzzles. Such puzzles gives us “typical” and characterization and “intuitive” understanding for some computational complexity classes.

On the other hand, there are some puzzles and games that give nontrivial interesting aspects of computational complexity classes. For example, let us consider ”14-15 puzzle” which is classic well known sliding puzzle. By parity, we can determine if one arrangement can be slid to the other in linear time. Moreover, we can always find a way for sliding between them in quadratic time. However, interestingly, finding the optimal solution is NP-complete in general. Through such classic puzzles, the reconfiguration problems are recently well investigated as a new framework of characterization of computational complexity classes. I give some recent results for these games and puzzles on graphs from the viewpoint of theoretical computer science.

SUMMARY: We present research on generalized janken. Janken is a very simple and well-known game in Japan. It is originated in China, and many variants are seen throughout the world. A variant of janken can be represented by an asymmetric digraph, where a vertex corresponds to a sign and an arc (x, y) indicates that sign x defeats sign y. However, not all asymmetric digraphs define useful janken variants, i.e., some janken variants may include a useless sign, which is strictly inferior to another sign in any case. We show research on janken variants in terms of useless signs, a measure of amusement, and ties between different signs.

SUMMARY: Delsarte (1973) gave the linear programming method to find bounds for the cardinality of codes with given distances. Delsarte’s method is useful to solve some optimization problems of maximizing the cardinality of a code in the Euclidean sphere or certain special association schemes, like Johnson schemes or Hamming schemes. A particular example of applications of the bound is the determination of the kissing number for a sphere. In this talk, we introduce an analogous theory of Delsarte’s method for regular graphs. This method gives bounds for the order of a regular graph with given eigenvalues. The optimal graphs for this bound are distance-regular graphs with high girth, like Moore graphs. We also introduce recent results on a generalization of this bound for regular uniform hypergraphs.

SUMMARY: We investigate a periodicity of Grover walk on complete graphs with self-loop at each vertex. We study an evolution matrix which steps forward a state of probability amplitude vector by using algebraic method. Then we find a periodic behaviour that the probability amplitude vector at each vertex gets back to initial state after some steps. It is shown that its fundamental period is \(2n\) for the walk which has \(n\) vertices.

SUMMARY: We discuss the difference of two mathematical models of data-diffusion; the deterministic and stochastic models. The deterministic model is given by a reaction-diffusion partial differential equation, and the stochastic model is given by a multi-dimensional jump Markov process. In this talk, by scaling the state and fluidizing data-pieces, we show that the difference of two models converges to 0 in probability on bounded time intervals by a law of large numbers.

SUMMARY: The Möbius type inclusion-exclusion integral is a representation of nonlinear integral with respect to nonadditive measures through the Möbius translation. We propose parameter estimation with backpropagation method of the Möbius type inclusion-exclusion integral mathematical model. Using this method, not only parameter determination but also data preprocessing can be performed automatically. We also attempt to interpret the sparse regularization of neural networks by representing this model as a discrete graph.

SUMMARY: In this talk, we analyze truncation errors in the evaluation of orthant probabilities of elliptical distributions. The procedure analyzed consists of repeated integration. The truncation error proliferate if some issues are left. Methods to settle these issues were proposed. Probabilities up to four dimensional cases, which can be evaluated precisely by another procedure, were evaluated using the proposed methods and compared. The result showed that proposed methods aids to reduce truncation error.

SUMMARY: In the fields of engineering and medicine, three-dimensional tissue (structure) is needed to be analyzed. There are CT and MRI equipment to analyze three-dimensional tissue, but they are expensive and have limited spatial resolution. On the other hand, although optical projection images such as a microscope are relatively inexpensive and have high spatial resolution, it is very difficult to reconstruct three-dimensional structure from the obtained image. In this talk, we treat structure of silicon gel and chromatin in the cell nucleus as an example. Using the topological information from the projection image of them, we acquire and analyze the three-dimensional information required for practical use. If this method can be arranged mathematically, we will be able to discover many applications in the future.

SUMMARY: When we consider a deformation of point cloud data, the deformation can be formalized as a sampled map, which is a finite subset of an underlying continuous map. In this talk, we provide a new construction of persistent homology for sampled maps, which can capture the homology induced map of the underlying map. The key idea is block matrix form of persistence modules on commutative ladders, and the functoriality of (1:3, 1:3) submatrices ensures the well-definedness of the persistent homology. Moreover, the functoriality can be generalized for arbitrary length and arbitrary diagonal submatrices, and enables to analyze 2-dimensional persistent homology.

SUMMARY: In the recent trend of extending discrete-to-continuum limit passages for gradient flows of single-species particle systems with singular and nonlocal interactions to particles of opposite sign, any annihilation effect of particles with opposite sign has been side-stepped. We present the first rigorous discrete-to-continuum limit passage which includes annihilation. This result paves the way to applications such as vortices, charged particles, and dislocations.

SUMMARY: A kind of generalization was proposed by the author for an integer-type algorithm for solving higher-order linear ODEs, which was proposed by the author and M. Hayashi several years ago, by means of algebraic extensions of the field of rational functions. By this generalization, for example, we can solve the higher-order linear ODEs whose coefficient functions are general algebraic functions, by means only of four arithmetical operations among integers. However, this generalization may cause extra solutions, because this generalization increases the orders of the ODEs. In this study, the author proposes a general method for removing these extra-solution components effectively from numerical results. Moreover, some successful numerical examples are given for the Schrodinger equations whose potential functions belong to the simple extension of the field of rational functions.

SUMMARY: We present an error analysis of piecewise linear nonconforming Crouzeix–Raviart finite element method (CR FEM) for the Poisson problem in 3dim without the shape regularity condition. First, we show that in this case, Crouzeix–Raviart finite element method is equivalent to the lowest order Raviart–Thomas finite element method (RT FEM). Next, using the Babu\(\rm {\breve {s}}\)ka–Aziz technique, we present error estimates of RT interpolation. Since RT FEM is confirming, a Céa type lemma is valid and error estimates of RT FEM are obtained from that of RT interpolation. Using the obtained equivalence of CR and RT FEMs, we finally present the targeted error estimates of CR FEM. We again emphasise that we do not impose the shape-regular condition for the mesh partition.

SUMMARY: For simulations of various problems, we need to consider complex boundary conditions. For example, boundary conditions involving a Laplace–Beltrami operator, such as a dynamic boundary condition and a generalized Robin condition play important roles for application to reduced-FSI model and Cahn–Hilliard equations. There are several works for error estimates of discontinuous Galerkin methods for a dynamic boundary condition and a generalized Robin condition. However, that study considers only a rectangle domain, so it is difficult to apply it to practical problems. In this study, we show the analysis and some numerical results of a discontinuous Galerkin method for Poisson equations with a generalized Robin boundary condition in a smooth domain.

SUMMARY: This paper proposes new finite element schemes of mass-lumping type for computing spherically symmetric solutions of multi-dimensional semilinear heat equations. We prove the positivity conservation property and error estimates. In particular, we are able to remove some obstructions in the standard finite element schemes. Moreover, the convergence of blow-up time is established.

SUMMARY: In an early paper (Contribution to the theory of upper and lower bounds in boundary value problems, J. Phys. Soc. Jpn., 1955, 10:1–8), based on the hypercircle-like method, H. Fujita proposed a novel method to provide pointwise error estimation for the boundary value problems. This paper is published in the dawning era of the finite element method (FEM), but it seems that the idea therein has never been applied to the error estimation for FEM. In this talk, we show the possibility to apply Fujita’s method to develop pointwise error estimation for FEM solutions to boundary value problems.

SUMMARY: We consider several approximate operators used in a particle method based on a Voronoi diagram. Under some assumptions on a weight function, we derive truncation error estimates for our approximate gradient and Laplace operators. Our results show that our approximate gradient and Laplace operators tend to the usual gradient and Laplace operators when the ratio (the radius of the interaction area/the radius of a Voronoi cell) is sufficiently large.

SUMMARY: We present the method of interval inclusion of optimal constants for second order error estimates of \(H_0^1\) function to its finite degree polynomial approximation.

SUMMARY: The method of fundamental solutions (MFS) is a mesh-free numerical solver for solving homogeneous linear partial differential equations, and it has been applied to solve Laplace equation, Helmholtz equation, modified Helmholtz equation, and so on. In this talk, we establish mathematical results on unique existence and exponential decay of approximation error of the MFS for Helmholtz-type equation.

SUMMARY: The scalar auxiliary variable (SAV) approach is a numerical method to solve gradient flows, which ensures some kind of stability by introducing a scalar auxiliary variable. This method is linear and unconditionally stable. In this study, we consider extending the SAV approach to Hamiltonian systems and propose a numerical scheme that is linear and energy conservative in a sense.

SUMMARY: The contents of three analytic methods, namely, analysis of periodic solutions in non-autonomous system, equilibrium points in autonomous system, and periodic solutions in autonomous system, are virtually identical. These individual method have been intended utilizing a new perspective algorithm for individual analytic method. Three individual programs are forming an unified program. The unified program has been made improving easy to grasp source code and to modify source code.

SUMMARY: We present a class of distributed delay differential equations that has periodic solutions of period 2, where the maximum delay of the equation is 1. The existence of the periodic solutions is proven, following the idea by Kaplan and Yorke (1974): an ansatz deduces a second order ordinary differential equation. We show that, for some special equations, there exist periodic solutions that can be expressed in terms of the Jacobi elliptic functions explicitly.

SUMMARY: We consider blow-up problem for delay differential equations with distributed delay. We discuss several types of distribution of the delay and show blow-up of solutions for the target problems.

SUMMARY: The Cauchy problem for a semilinear second order differential equation is considered. The effects of the spatial expansion and contraction are studied through the problem.

SUMMARY: Global solutions for the semilinear complex Ginzburg–Landau type equation are considered in homogeneous and isotropic spaces. The Asymptotic behaviors of the solutions are also considered.

SUMMARY: In this talk, we consider the dynamics of a 2-pulse solution arising in a bistable three-component reaction-diffusion system, for which two stable pulse solutions are glued together and bounded in a finite region. First, we numerically investigate the PDE system to explore the dynamics of the 2-pulse solution, and then clarify their mechanism, especially the bifurcation structure behind them, by means of seven-dimensional ODEs which describe the interface motions of the 2-pulse solution.

SUMMARY: In this talk, we treat an interface equation related to some mathematical models of material science. In particular, we show the existence of traveling wave solutions of it, that is, planer wave solution, V-shaped traveling wave solution, inverse U-shaped traveling wave solution. Moreover, it is shown that the solution to the interface equation with prescribed contact angle eventually converges to a portion of one of such three traveling waves.

SUMMARY: Report that the simple but robust mathematical principle of so-called “Turing instability” is important also here as a design manner, which can be a useful principle to understand a part of the rationality of a kind of life creature’s activities in post-transcrition-translation process since Professor Alan Turing wrote his very famous paper. Terrestrial cyanobacteria of Nostochineae, while unicellular, makes a group of population, adapts to the environment and makes various efforts to survive. For example, when performing Biological Nitrogen Fixation (BNF), some of them will make differentiation into “heterocysts” cells, if nitrogen compounds feel poor in themselves. These are cells specialized for the above BNF, wherein cyanobacteria protects nitrogenase, which is an enzyme protein of the above reaction, from oxygen. They make BNF in those cells, as pattern formation is accompanied. We have a fundamental interest of its mathematical mechanism.

SUMMARY: The Leaky Integrate and Fire (LIF) model is widely used to describe the dynamics of neural networks. We can derive the Fokker–Planck equation from the LIF model after a diffusive approximation of the mean-field limit of a stochastic differential equation system. In this talk, we show that the Fokker–Planck equation generates a spatially nonconstant solution with a time-periodical motion induced by a delay in the effect of the total activity on the neurons.

SUMMARY: Let G be an eulerian. Then each 2-chain cover in G generates distinct Euler tours in G.

SUMMARY: This talk is on my result in 2013 about when a strict inequality of Kazhdan–Lusztig polynomials such as \(P_{uw}(1)>P_{vw}(1)\) hold in a crystallographic Coxeter system \(W\). I will show how to prove it with Deodhar inequality and certain positivity of \(R\)-polynomials at \(q=1\).

SUMMARY: I will present my two formulas on the bigrassmannian statistic, certain enumeration of bigrassmannian permutations, as interactions of matrix, lattice, and Coxeter group theories. The first is over the set of permutations under Bruhat order while the second is over alternating sign matrices as a natural generalization of the first. We can interpret both formulas as weighted counting of the classical inversion statistic.

SUMMARY: This study investigates the unitary equivalence classes of translation-invariant two-dimensional two-state quantum walks. We show that unitary equivalence classes of such quantum walks are essentially parameterized by two real parameters.

SUMMARY: We consider a quantum search problem on graphs using a type of continuous time quantum walk. In this talk, we show a scheme to determine the effectiveness of the search on given graphs. The scheme consists of equitable partition of given graphs and the framework of perfect state transfer problem. By combining the two methods and spectral decomposition technique, we give a condition to determine the effectiveness of the search.

SUMMARY: A quantum walk has been studied as a quantum analogue of a classical random walk. One of the most effective applications of quantum walks is searching probrem. It is known that quantum walks can propose more efficient system to find a target in a graph than that given by classical random walks. In this talk, we present a quantum walk on edge signed graphs in which every edge is declared positive or negative. In addition, we describe qunatum searching model which efficiently find a negatively signed edge on an edge signed complete graph.

SUMMARY: We propose a quantum walk defined by digraphs (mixed graphs). This is essentially a special case of twisted Szegedy walks, and is equipped with a 1-form function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give a consideration by computer on the identification of digraphs by their eigenvalues.

SUMMARY: We consider stationary measures of quantum walks on a half line determined by singular continuous measures, in particular, the Riesz and Cantor measures, respectively. Our approach is based on mainly transfer matrix method.

SUMMARY: It is well known that the spectrum of the Hadamard walk in one dimension is just continuous one. In this talk, we discuss the existence of point spectrum for the Hadamard walk with defects.

SUMMARY: Quantum walks were proposed as quantum mechanical counterparts of random walks and expected to realize topological phenomena. Here we consider position depending split-step quantum walks on cycles defined as a finite version of Kitagawa’s model, which include two-phase quantum walks given by Balu et al. In this talk, we show the necessary and sufficient condition that the birth eigenspace is nontrivial. Moreover, we investigate an evaluation of the long-time behavior of the two-phase quantum walk by using a spectral mapping theorem.

SUMMARY: Quantum walks have been intensively studied as quantum versions of classical random walks. The time evolution of the quantum walk is defined by a unitary operator, which is expressed as a product of coin and shift operators. In this study, we analyze a one-defected quantum walk on the multi-dimensional lattice by using the quantum walk version of spectral mapping theorem. We get some different results from the previous study by Fuda et al.

SUMMARY: On Prime Number, we propose New Twin Prime Number Concept and have gotten some numbers named as TTTTTTwins prime number sets. That is defined as 6Twin prime numbers. These TTTTTT mean no prime between T and T. (NOT TTpTTTT or NOT TTTTpTT etc.) and 6 continued Twins. One of these sets of prime is [[10099096127, 10099096129], [10099096169, 10099096171], [10099096307, 10099096309], [10099096349, 10099096351], [10099096361, 10099096363], [10099096367, 10099096369]] by Maple PG. We have 4sets of TTTTTTwins now. And more, we present Some New Prime Number sets which are calucurated by Maple PG.

SUMMARY: Let \(G\) be a finite connected simple graph and \(P_G\) be the edge polytope of \(G\). The combinatorial structure of \(P_G\), especially the types of triangulations that \(P_G\) admits, is an interesting problem, which has been studied extensively. A necessary and sufficient condition for \(P_G\) to possess a regular unimodular triangulation is obtained. However, this condition is not easy to apply to a given graph by merely inspecting the graph. Then, for a graph \(G\), we will obtain several criteria for the existence of a regular unimodular triangulation of \(P_G\) in terms of simple data related to the graph.

SUMMARY: We have obtained the best constant of discrete Sobolev inequality corresponding to Hamilton path on the regular polyhedra. Let \(N\) be the number of vertices. We introduce the discrete Laplacian which is \(N\times N\) real symmetric matrix. The discrete Laplacian has an eigenvalue \(0\) whose corresponding eigenspace is 1 dimension. If we introduce the pseudo Green matrix, then the matrix is reproducing kernel by setting appropriate vector space and inner product. The maximum of the diagonal values of the pseudo Green matrix is the best constant of this discrete Sobolev inequality.

SUMMARY: The topological complexity is a numerical invariant closely related to robot motion planning. If the motion is required certain symmetricity, the symmetric topological complexity plays an important role to design algorisms. This talk will present a combinatorial approximation of symmetric topological complexity using a categorical model of configuration space.

SUMMARY: Pseudo-randomness of (di)graphs is a measure of ^^ ^^ randomness” of given deterministic (di)graphs. This has been widely applied to various problems in graph theory and combinatorial optimization. For example, Alon (SIAM J. Discrete Math., 2006) proved that the feedback arc set problem for tournaments is NP-hard by focusing on ranking tournaments and pseudo-randomness of Paley tournaments.
In this talk, we give a generalized version of the Alon’s proof by focusing on a tight relationship between the pseudo-random property and (di)graph spectra.

SUMMARY: The number of flats of the braid arrangement, a typical hyperplane arrangement, coincides with the Bell number and the Stirling number of the second kind. In this talk, we enumerate the flats of the extended Catalan and Shi arrangements, which are deformations of the braid arrangements.

SUMMARY: Let us consider the forests consisting \(k\) connected components which are subgraphs of the complete bipartite graph. For these forests, consider the generating function. We show that the Hessian of the generating function does not vanish by calculating the eigenvalues of the Hessian matrix of the generating function.

SUMMARY: We define a new weighted Ihara zeta function and a new weighted Ihara \(L\)-function of a graph \(G\), and present their determinant expressions. As a corollary, we present a decomposition formula for the new weighted Ihara zeta function of a regular covering of \(G\) by its new weighted Ihara \(L\)-functions. As applications, we give the spectrum of the transition probability matrices of non-backtracking random walks for regular graphs and semiregular bipartite graphs.

SUMMARY: We consider the spectrum of a 1-dimensional 2-state non-unitary quantum walk which is defined by Mochizuki, Kim and Obuse. This time-evolution operator acts on the Hilbert space \(\ell ^2(\mathbb {Z};\mathbb {C}^2).\) This model depends on a parameter \(\gamma >0.\) The loss and gain of the photon is controlled by \(\gamma \) in this model. If \(\gamma =0\), then the photon energy does not either lose or gain. In particular, the model is a unitary operator in the case of \(\gamma =0.\) However, in the case of \(\gamma >0\), the model is not unitary. Moreover, it is not a normal operator. This fact means that we can not use some general theorems of the unitary operator when we analyze some properties of this model. I this talk, we classify the spectrum of the model into three different cases according to the size of \(\gamma .\)

SUMMARY: In this talk, we consider the spectral mapping theorem of an abstract non-unitary quantum walk. In particular, we focus on the continuous spectrum of the time evolution since we already have got the result for the point spectrum of it. Our time evolution is not always normal, so that we can not use some general theorems for the spectral analysis. We also talk about the relation between our model and the Ihara zeta function, which has been intensively investigated in the graph theory.

SUMMARY: Some recent results on safe set problems in vertex-weighted graphs will be reviewed.

SUMMARY: Let \(G\) be a graph. A mapping \(c: E(G) \rightarrow \mathbb N\) is called an edge-coloring of \(G\) and \(c(e)\) is called the color of an edge \(e\). A graph with an edge-coloring map is called an edge-colored graph and denoted by \((G,c)\). A subgraph \(H\) of \(G\) is called rainbow if every pair of edges in \(H\) have distinct colors and \(H\) is said to be properly colored, or shortly PC, if any two adjacent edges have different colors.

In this talk, first we will consider structures of edge-colored complete bipartite graphs \((K_{n,m},c)\) without PC cycles of length four, and next the number of disjoint PC cycles in \((K_{n,m},c)\) will be discussed related with Bermond–Thomassen Conjecture. Finally several problems and results around this topic will be given.

SUMMARY: A cycle of order \(k\) is called a \(k\)-cycle. A non-induced cycle is called a chorded cycle. A graph \(G\) of order \(n\ge 4\) is chorded pancyclic if \(G\) contains a chorded \(k\)-cycle for every integer \(k\) with \(4\le k\le n\). Cream, Gould and Hirohata (2017) conjectured that a hamiltonian graph \(G\) of order \(n\ge 4\) satisfying \(|E(G)|\ge \frac {1}{4}n^2\) is chorded pancyclic unless \(G\) is either \(K_{\frac {n}{2},\frac {n}{2}}\) or \(K_3\times K_2\). In this talk, we affirmatively answer this conjecture by showing that if a graph \(G\) of order \(n\) with \(|E(G)|\ge \frac {1}{4}n^2\) contains a \(k\)-cycle, then \(G\) contains a chorded \(k\)-cycle, unless \(k=4\) and \(G\) is either \(K_{\frac {n}{2},\frac {n}{2}}\) or \(K_3\times K_2\), Then observing that \(K_{\frac {n}{2},\frac {n}{2}}\) and \(K_3\times K_2\) are exceptions only for \(k=4\), we further relax the density ondition for sufficiently large \(k\).

SUMMARY: A proper \(n\)-coloring \(c : V(G) \to \{1, \ldots , n\}\) of a graph \(G\) is a complete \(n\)-coloring if every pair of colors appears on at least one edge. The maximum number \(n\) such that \(G\) has a complete \(n\)-coloring is called the achromatic number of \(G\). A graph \(G\) is locally connected if the neighbourhood of every vertex of \(G\) is connected. In this talk, we shall show that both the chromatic and the achromatic numbers of connected and locally connected graph \(G\) are exactly \(3\) if and only if \(G\) is isomorphic to a complete tripartite graph.

SUMMARY: For positive integers \(t\) and \(n\), a facial \(t\)-complete \(n\)-coloring of a graph \(G\) embedded on a closed surface is a color assignment \(c:V(G)\to \{1,\ldots , n \}\) of the vertices such that any \(t\)-tuple of colors appears on the boundary of some face of \(G\). The facial \(t\)-achromatic number of \(G\), denoted by \(\psi _t(G)\), is the maximum number \(n\) such that \(G\) has a facial \(t\)-complete \(n\)-coloring. The facial \(t\)-achromatic number depends on the embedding of \(G\) in genetral. That is, for another embedding \(f(G)\) of \(G\), \(\psi _t(f(G))\) may not be equal to \(\psi _t(G)\). In this talk, we evaluate the difference between \(\psi _3(G)\) of a triangulation \(G\) on a closed surface and \(\psi _3(f(G))\) of another triangulation of \(G\).

SUMMARY: In this talk, we show the non-\(1\)-planarity of the lexicographic product of a theta graph and \(K_2\). This result completes the proof of the conjecture that a graph \(G\circ K_2\) is \(1\)-planar if and only if \(G\) has no edge belonging to two cycles.

SUMMARY: In 1986, Stanley introduced two classes of lattice polytopes associated to finite partially ordered sets, which are called order polytopes and chain polytopes. It is known that the Ehrhart polynomials of these polytopes coincide with some counting polynomials of \(P\)-partitions. In this talk, from the theory of (left) enriched \(P\)-partitions, which are introduced and studied by Stembridge and Petersen, we introduce enriched order polytopes and enriched chain polytopes. In particular, we show that the Ehrhart polynomials of these polytopes coincide with some counting functions of left enriched \(P\)-partitions.

SUMMARY: A finite subset \(X\) of the Euclidean space is called an \(s\)-distance set if the number of the distances of two distinct vectors in \(X\) is equal to \(s\). An \(s\)-distance set \(X\) is said to be maximal if any vector cannot be added to \(X\) while maintaining \(s\)-distance. We investigate a necessary and sufficient condition for vectors to be added to the regular simplex such that the set has only \(2\) distances. We construct several maximal \(2\)-distance sets that contain the regular simplex.