Louigi Addario-Berry, Nicolas Broutin, Gabor Lugosi
We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α^* \log n$ edges where α^* ~ 3.5911 is the unique solution of the equation $alpha log(alpha) - α=1. This answers a question posed by Janson (1999).
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For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model. We show that several results concerning the length of the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$. For a constant $c$, we show that asymptotically almost surely the length of the longest path is at least $(1-(1+ε(c))ce^{-c})n$ for some function $ε(c)\to 0$ as $c\to \infty$, a…
Chords of longest cycles in graphs with large circumferences
本文证明了对于具有大周长的图，Thomassen 和 Harvey 关于最长圈弦的猜想均成立，并推广至包含线性森林的最长圈情形。
We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random DAG. This completes the study of natural distances in random DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also obtain large deviation bounds for the minimum of a branching r…
Improved asymptotic upper bounds for the minimum number of pairwise distinct longest cycles in regular graphs
本文改进正则哈密顿图中两两不同最长圈数量的渐近上界，证明 Haythorpe 猜想在渐近意义上不成立，并证明非哈密顿 2-连通正则图的该不变量可被常数界定。
Two sharp lower bounds for the length of a longest cycle $C$ of a graph $G$ are presented in terms of the lengths of a longest path and a longest cycle of $G-C$, denoted by $\overline{p}$ and $\overline{c}$, respectively, combined with minimum degree $δ$: (1) $|C|\geq(\overline{p}+2)(δ-\overline{p})$ and (2) $|C|\geq(\overline{c}+1)(δ-\overline{c}+1)$.
Chords of longest cycles passing through a specified small set
本文研究经过指定小集合的最长圈的弦，推广Thomassen和Zhang的相关结果，证明了2-连通立方图、最小度≥4的3-连通图及3-连通平面图中满足特定条件的最长圈必含弦，并扩展了Li和Liu的结果。
The longest-edge bisection algorithm may produce degenerating tetrahedra
该研究通过构造显式序列证明最长边平分算法会产生退化四面体，违反形状正则性与角条件，说明最长边平局破局无法保证非退化。
Let $lct(G)$ be the minimum cardinality of a set of vertices that intersects every longest cycle of a 2-connected graph $G$. We show that $lct(G)\leq k-1$ if $G$ is a partial $k$-tree and that $lct(G)\leq \max \{1, {ω(G){-}3}\}$ if $G$ is chordal, where $ω(G)$ is the cardinality of a maximum clique in $G$. Those results imply that all longest cycles intersect in 2-connected series parallel graphs and in 3-trees.
We consider a variant of the continuous and discrete Ulam-Hammersley problems: we study the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path.For both cases (continuous and discrete) our approach rely on couplings with well-studied models: respectively the classical Ulam-Hammersley problem and last-passage percolation with geometric weights. Thanks to these couplings we obtain explicit limiting shapes in both settings.We also es…
Let $\lpt(G)$ be the minimum cardinality of a set of vertices that intersects all longest paths in a graph $G$. Let $ω(G)$ be the size of a maximum clique in $G$, and $\tw(G)$ be the treewidth of $G$. We prove that $ \lpt(G) \leq \max\{1,ω(G)-2\}$ when $G$ is a connected chordal graph; that $\lpt(G) =1$ when $G$ is a connected bipartite permutation graph or a connected full substar graph; and that $\lpt(G) \leq \tw(G)$ for any connected graph $G$.
Longest cycles and longest chordless cycles in $2$-connected graphs
本文研究了2连通图中最长环与无弦环的关系，提出猜想等价于Harvey猜想，并证明该猜想在特定图类中成立。
Longest cycles and Dirac-type results in highly connected graphs
本文证明了Bondy关于k连通图中最长圈外路径长度不超过k-1的猜想对所有足够大的图成立，并给出了一个Dirac型定理以给出k连通图中最长圈长度的下界。
We consider the likely size of the endpoint sets produced by Posa rotations, when applied to a longest path in a random graph with $cn,\,c\geq 2.7$ edges that is conditioned to have minimum degree at least three.
The detour covering number and cummerbund covering number of a graph
本文提出图的绕行覆盖数与腰带覆盖数等新概念，并研究最小度、禁用子图、连通度、围长等条件下图的覆盖性质，确定2-连通二部图的最小腰带覆盖数及极图。
Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that lpt(G) \leq ceiling(n/4-n^{2/3}/90), if G is connected, lct(G) \leq ceiling(n/3-n^{2/3}/36), if G is 2-connected, and \lpt(G) \leq 3, if G is a connected circular arc graph. Our bound on lct(G) improves an earlier result of Thomassen and our bound for circular arc graphs relates to an earlier statement of Balister \emph{et al.} the argument of which contains a gap. Furthermore, we p…
In this paper, we continue the study of the Hamiltonian and longest $(s, t)$-paths of supergrid graphs. The Hamiltonian $(s, t)$-path of a graph is a Hamiltonian path between any two given vertices $s$ and $t$ in the graph, and the longest $(s, t)$-path is a simple path with the maximum number of vertices from $s$ to $t$ in the graph. A graph holds Hamiltonian connected property if it contains a Hamiltonian $(s, t)$-path. These two problems are well-known NP-complete for general supergrid graphs. An $O$-shaped supergrid graph is a special kind of a rectangular grid graph with a rectangular hol…
Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $δ$ and $p$ - the order of a longest path in $G$. This is a common generalization of two earlier classical results for 2-connected graphs due to Dirac: (i) $c\ge \min\{n,2δ\}$; and (ii) $c\ge\sqrt{2p}$. Moreover, the result is stronger than (ii).
We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree $δ$ with $δ\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle unless $G$ belongs to an easily specified class of graphs with $κ(G)=2$ and $τ(G)=1$. The second result follows immediately from the first result: (ii) if $G$ is a 3-connected 1-tough graph with $δ\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.
Every 4-connected graph $G$ with minimum degree $δ$ and connectivity $κ$ either contains a cycle of length at least $4δ-κ-4$ or every longest cycle in $G$ is a dominating cycle.
Supergrid graphs contain grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for general supergrid graphs were known to be NP-complete. A graph is called Hamiltonian if it contains a Hamiltonian cycle, and is said to be Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it. In this paper, we first prove that every L-shaped supergrid graph always contains a Hamiltonian cycle except one trivial condition. We then verify the Hamiltonian connectivity of L-shaped supergrid graphs except few conditions. The Ham…
We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form. The direct form extends to other subgraphs and other edge-weight distributions. We illustrate the clique results for various val…
Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput, and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all hamiltonian 4-regular graphs of suffi…
We study the minimum total weight of a disk triangulation using vertices out of $\{1,\ldots,n\}$, where the boundary is the triangle $(123)$ and the $\binom{n}3$ triangles have independent weights, e.g. $\mathrm{Exp}(1)$ or $\mathrm{U}(0,1)$. We show that for explicit constants $c_1,c_2>0$, this minimum is $c_1 \frac{\log n}{\sqrt n} + c_2 \frac{\log\log n}{\sqrt n} + \frac{Y_n}{\sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $\frac14\log n + O_P(\sqrt{\log n}) $ vertices. Moreover, for disk triangulations that are canonical, in that n…
The minimum number of detours in graphs
本文研究图中最长路径（绕行路径）的最小数量，证明在最小度至少为2且阶至少为9的连通图中，若绕行路径数为奇数则至少为9，并排除3、5、7三种可能，同时提出两个开放问题。
Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hatα\log n$ edges where $\hatα$ is the unique solution of the equation $α\log(\frac{d-2}{d-1}α) - α= \frac{d-3}{d-2}$, for $α> \frac{d-1}{d-2}$.
Let $G$ be a $t$-tough graph of order $n$ and minimum degree $δ$ with $t>1$. It is proved that if $δ\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.
Erdős determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized. Erdős' work and gave the maximum size $h(n,c,k)$ of graphs with prescribed order $n$, circumference $c$ and minimum degree at least $k.$ But for some triples $n,c,k,$ the maximum size is not attained by a graph of minimum degree $k.$ For example, $h(15,14,3)=77$ is attained by a unique graph of minimum degree $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a graph with prescribed order, circumfer…
We present eighteen exact analogs of six well-known fundamental Theorems (due to Dirac, Nash-Williams and Jung) in hamiltonian graph theory providing alternative compositions of graph invariants. In Theorems 1-3 we give three lower bounds for the length of a longest cycle $C$ of a graph $G$ in terms of minimum degree $δ$, connectivity $κ$ and parameters $\bar{p}$, $\bar{c}$ - the lengths of a longest path and longest cycle in $G\backslash C$, respectively. These bounds have no analogs in the area involving $\bar{p}$ and $\bar{c}$ as parameters. In Theorems 11 and 12 we give two Dirac-type resu…
The minimum number of detours in a connected graph of minimum degree three
本文研究了具有最小度数三的连通图中最少迂回数的问题，证明了a(3,n)=36(n≥18)并给出了其他相关结果。
For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent vertices, then $c\ge p-1$. This improves results of Nash-Williams (in terms of minimum degree $δ$ and order $n$), Bondy (in terms of degree sum $σ_{3}$ and order $n$), and Enomoto, Heuvel, Kaneko and Saito (in terms of degree sum $σ_3$, order $n$ and relative length $diff(G)=p-c$).
On certain blocking sets and the minimum weight of the code of generalised polygons
本文通过刻画广义多边形中的阻塞集，推导出关联矩阵生成的码在最小重量及最小重量码字方面的新结果。
A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the "yea" voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. Freixas and Molinero have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.
A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates is not larger than 1. The Hamiltonian path (cycle) problem is to determine whether a graph contains a simple path (cycle) in which each vertex of the graph appears exactly once. This problem is NP-complete for general graphs and it is also NP-complete for general supergrid graphs. Despite the many applications of the problem, it is still open for many classes…
Odd and even cycle lengths, minimum degree and chromatic number in graphs
研究给定图中奇偶数圈长度情况下团数与色数的关系，通过证明和构造图得出相关结论，如给定奇数圈长度条件时的色数界限及特殊图类，同理对偶数圈长度也有研究，综合结果可推导多种情况下团数与色数的关系。
This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players …
Small hitting sets for longest paths and cycles
该研究受经典图论问题与猜想启发，运用结构图论方法，改进了图中击中所有最长路径、最长环的最小顶点集规模的相关界，并推进了顶点传递图中路径与环最大长度的猜想研究。
The minimum weight of the code of intersecting lines in ${\rm PG}(3,q)$
本文研究了${\rm PG}(3,q)$中相交线码的最小重量及偶数情况下第二小重量码的性质。
On the Random Minimum Spanning Subgraph Problem for Hypergraphs
本文研究了超图的随机最小生成子图问题，将完全加权图中最小生成树权重趋于分布相关常数的经典结果推广至超图环境。
We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimu…
Towards the characterization of minimum weight codewords of Schubert codes
本文验证了Schubert码最小重量码字猜想，证明了Schubert分解可码字在有限多个例外情况下成立。
For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis [Inf. Process. Lett. 110 (2010) 970-974 ]. In this work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth two) which is a superclass of outerplanar graphs.
Hitting all longest paths in $H$-free graphs and $H$-graphs
该研究提出结构定理细化最长路径横贯，给出无$P_t$图、无（bull, chair）图等遗传图类的最长路径横贯数常数上界，并证明连通$H$-图的最长路径横贯数存在仅依赖$H$的常数上界。
Digraphs of potential barriers: properties of their tree structure and algorithm for constructing minimum spanning forests
针对弧权由带环无向图P导出的无环加权有向图V，研究其性质并提出构造最小权有向树和最小权生成有向森林的高效算法。
There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let $G$ be a graph with minimum degree at least $k+1$. We prove that if $G$ is bipartite, then there are $k$ cycles in $G$ whose lengths form an arithmetic progression with common difference two. For general graph $G$, we show that $G$ contains $\lfloor k/2\rfloor$ cycles with consecutive even lengths and $k-3$ cycles whose lengths form an arithmetic progression with common difference one or two. In addition, if $G$ is 2-connected and non-…
Algorithm for Constructing Related Spanning Directed Forests of Minimum Weight
提出一种构造最小权有向生成森林的算法，可在树数变化时保持最小森林间的最大亲和度，复杂度对稠密图不超过 $O(N^3)$，并输出所有允许 $k$ 下由 $k$ 棵树组成的相关最小森林。
We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^γ$ where $γ\leq 1$ and $U$ is the uniform $[0,1]$ random variable. There is also a constraint that the spanning tree $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,γ$, the asymptotic value of the optimum weight via the consideration of a dual problem.
In the 1960s, Erdős and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erdős-Gallai conjecture for random graphs and for graphs with linear minimum degree.
Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.
The second minimum weight of Grassmann codes
本研究通过格拉斯曼簇的特殊分解，独立证明了格拉斯曼码最小距离的Nogin定理，并运用相同思路计算出其第二最小重量。
On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes
本文通过另一种证明刻画任意阶射影 Reed-Muller 码的最小距离与最小重量码字，给出其数量并讨论维数公式的等价性。
In this paper we investigate the number of minimum weight codewords of some dual Algebraic-Geometric codes associated with the Giulietti-Korchmáros maximal curve, by computing the maximal number of intersections between the Giulietti-Korchmáros curve and lines, plane conics and plane cubics.
In a recent paper, Beniamini and Nisan gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph $G \subseteq K_{n,n}$ has a perfect matching, together with an efficient algorithm for computing the coefficients of the monomials of this polynomial. We give the following generalization: Given an arbitrary non-negative weight function $w$ on the edges of $K_{n,n}$, consider its set of minimum weight perfect matchings. We give the real multilinear polynomial for the Boolean function which determines if a graph $G \subseteq…
The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a m…
In this note, we construct new doubly even self-dual codes having minimum weight $20$ for lengths $112$, $120$ and $128$. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length $112$.
We find out the number of different partitions of an n-kilogram stone into the minimum number of parts so that all integral weights from 1 to n kilograms can be weighed in one weighing using the parts of any of the partitions on a two-pan balance. In comparison to the traditional partitions, these partitions have advantage where there is a constraint on total weight of a set and the number of parts in the partition. They may have uses in determining the optimal size and number of weights and denominations of notes and coins.
The Minimum Weighting Ratio Problem and Its Application in Chordal Graphs
本文研究了边加权连通图G的生成树T与G的加权求和之比的最小化问题，提出定理将其简化并证明可在多项式时间内求解，同时将该最优问题应用于弦图。
It is shown that the residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$ (resp.\ $24k+8$) and minimum Lee weight $8k+4 \text{ or }8k+2$ (resp.\ $8k+8 \text{ or }8k+6$) is a binary extremal doubly even self-dual code for every positive integer $k$. A number of new self-dual $\mathbb{Z}_4$-codes of length $24$ and minimum Lee weight $10$ are constructed using the above characterization. These codes are Type I $\mathbb{Z}_4$-codes having the largest minimum Lee weight and the largest Euclidean weight among all Type I $\mathbb{Z}_4$-codes of that length. In addition, new extremal Type …
We study the minimum spanning arborescence problem on the complete digraph $\vec{K}_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent uniform random variable $U^α$ where $α\leq 1$ and $U$ is uniform $[0,1]$. There is also a constraint that the spanning arborescence $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,α$, the asymptotic value of the optimum weight via the consideration of a dual problem.
In this paper, we give a new upper bound on the minimum Euclidean weight of Type II $\ZZ_{2k}$-codes and the concept of extremality for the Euclidean weights when $k=3,4,5,6$. Together with the known result, we demonstrate that there is an extremal Type II $\ZZ_{2k}$-code of length $8m$ $(m \le 8)$ when $k=3,4,5,6$.
Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~…
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