Related papers: Bisections of graphs
A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results…
A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and…
For an integer $k\ge 2$, let $G$ be a graph with $m$ edges and without cycles of length $2k$. The pivotal Alon-Krivelevich-Sudakov Theorem on Max-Cuts states that $G$ has a bipartite subgraph with at least $m/2+\Omega(m^{(2k+1)/(2k+2)})$…
In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 -…
The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over…
Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider…
In this paper, motivated by a problem of Scott and a conjecture of Lee, Loh and Sudakov we consider bisections of directed graphs. We prove that every directed graph with $m$ arcs and minimum semidegree at least $d$ admits a bisection in…
Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. We are interested in giving…
Consider two graphs G_1 and G_2 on the same vertex set V and suppose that G_i has m_i edges. Then there is a bipartition of V into two classes A and B so that for both i=1,2 the number of edges between A and B in G_i is (1+o(1))m_i/2. This…
The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation…
Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class…
In a graph $G=(V,E)$, a bisection $(X,Y)$ is a partition of $V$ into sets $X$ and $Y$ such that $|X|\le |Y|\le |X|+1$. The size of $(X,Y)$ is the number of edges between $X$ and $Y$. In the Max Bisection problem we are given a graph…
A celebrated result of Alon from 1993 states that any $d$-regular graph on $n$ vertices (where $d=O(n^{1/9})$) has a bisection with at most $\frac{dn}{2}(\frac{1}{2}-\Omega(\frac{1}{\sqrt{d}}))$ edges, and this is optimal. Recently, this…
We study the cutwidth measure on graphs and ways to bound the cutwidth of a graph by partitioning its vertices. We consider bounds expressed as a function of two quantities: on the one hand, the maximal cutwidth y of the subgraphs induced…
The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In…
The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…
For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…
Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise…
A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…
We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an…