Related papers: Heavy and Light Paths and Hamilton Cycles

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…

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…

In this paper, we study long paths and Hamiltonian paths in inhomogenous random graphs. In the first part of the paper, we consider an inhomogenous Erd\H{o}s-R\'enyi random graph $G_E$ with average edge density $p_n.$ We prove that if…

Let $G$ be a graph on $n\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced…

Let $c(G)$ denote the circumference of a graph $G$, i.e., the number of vertices in its longest cycle. For positive integers $n$ and $k$ with $n>k$, let $\varGamma(n;k)$ be the class of graphs of order $n$ with $c(G) = n-k$ such that every…

A heavy path in a weighted graph represents a notion of connectivity and ordering that goes beyond two nodes. The heaviest path of length l in the graph, simply means a sequence of nodes with edges between them, such that the sum of edge…

The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single…

Consider a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$. The weight of the shortest (minimum-weight) path $P_1$ between two given vertices is known to be $\ln n / n$, asymptotically. Define…

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…

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random…

A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order…

A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the…

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k +…

It is proved that if $G$ is a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$ then either $G$ has a cycle of length at least $\min\{n,2\delta+4\}$ or $G$ is the Petersen graph.

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…

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…

If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $\delta n$ edges, where $\delta> 1/2$. Let $\Psi(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic…

In graph theory, the longest path problem is the problem of finding a simple path of maximum length in a given graph. For some small classes of graphs, the problem can be solved in polynomial time [2, 4], but it remains NP-hard on general…