A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any .     

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single time (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y ^–2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.     

2-阶邻域连通无爪图的Hamilton性

设Ｇ是无爪图．对ｘ∈Ｖ（Ｇ），若Ｇ［Ｎ（ｘ）］不连通，则存在ｙｉ∈Ｖ（Ｇ）－｛ｘ｝（ｉ－１，２），使｜Ｎ（ｙｉ）∩Ｋｉ（ｘ）｜≥２，且｜Ｎ（ｙｉ）∩Ｎ（Ｋｉ＋１（ｘ））｛ｘ｝｜≥２（ｉ模２），那么称无爪图Ｇ是强２－阶邻域连通的，其中Ｋ１（ｘ），Ｋ２（ｘ）分别表示Ｇ［Ｎ（ｘ）］的两个分支．本文证明了：连通且强２－阶邻域连通的无爪图是Ｈａｍｉｌｔｏｎ图．     

Derivation of the probability distribution functions for succession quota random variables

The probability distribution functions (pdf's) of the sooner and later waiting time random variables (rv's) for the succession quota problem (k successes and r failures) are derived presently in the case of a binary sequence of order k. The probability generating functions (pgf's) of the above rv's are then obtained directly from their pdf's. In the case of independent Bernoulli trials, expressions for the pdf's in terms of binomial coefficients are also established.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

Generalizations of tournaments: A survey

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly nontrivial, even for these “tournament-like” digraphs. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 171–202, 1998     

清理三角债问题的一种方法

本文给出了一种清理三角债问题中的资金分配方法，用以实现用有限的资金清理最大数额债务的目的．由于该法是在网络图上进行的，较之以往的方法具有直观、操作简单的优点．     

Relative Length of Longest Paths and Cycles in Graphs

For a graph G, let diff(G) = p(G) − c(G), where p(G) and c(G) denote the orders of a longest path and a longest cycle in G, respectively. Let G be a 3-connected graph of order n. In the paper, we give a best-possible lower bound to σ₄(G) to assure diff(G) ≤ 1. The result settles a conjecture in J. Graph Theory 37 (2001), 137–156.     

Understanding chicken walks on n × n grid: Hamiltonian paths,discrete dynamics,and rectifiable paths

Understanding animal movements and modeling the routes they travel can be essential in studies of pathogen transmission dynamics. Pathogen biology is also of crucial importance, defining the manner in which infectious agents are transmitted. In this article, we investigate animal movement with relevance to pathogen transmission by physical rather than airborne contact, using the domestic chicken and its protozoan parasite Eimeria as an example. We have obtained a configuration for the maximum possible distance that a chicken can walk through straight and nonoverlapping paths (defined in this paper) on square grid graphs. We have obtained preliminary results for such walks which can be practically adopted and tested as a foundation to improve understanding of nonairborne pathogen transmission. Linking individual nonoverlapping walks within a grid‐delineated area can be used to support modeling of the frequently repetitive, overlapping walks characteristic of the domestic chicken, providing a framework to model fecal deposition and subsequent parasite dissemination by fecal/host contact. We also pose an open problem on multiple walks on finite grid graphs. These results grew from biological insights and have potential applications. © 2014 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.     

On relative length of longest paths and cycles

For a graph G, p(G) and c(G) denote the order of a longest path and a longest cycle of G, respectively. In this paper, we prove that if G is a 3 ‐connected graph of order n such that the minimum degree sum of four independent vertices is at least n+ 6, then p(G)?c(G)?2. By considering our result and the results in [J Graph Theory 20 (1995), 213–225; Amer Math Monthly 67 (1950), 55], we propose a conjecture which is a generalization of Bondy's conjecture. Furthermore, using our result, for a graph satisfying the above conditions, we obtain a new lower bound of the circumference and establish Thomassen's conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 279–291, 2009