Related papers: A graph partition problem

Given a symmetric D*D matrix M over {0,1,*}, a list M-partition of a graph G is a partition of G's vertices into D parts associated with the rows of M. The part of each vertex is chosen from a given list so that no edge of G maps to a 0 in…

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that…

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that…

In a graph $G$, let $\mu_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $\lambda K_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and \[ \mu_G(xy)=\begin{cases} \lambda &\mbox{if $x\in U$ and $y\in V$ or if…

A $(\delta\geq k_1,\delta\geq k_2)$-partition of a graph $G$ is a vertex-partition $(V_1,V_2)$ of $G$ satisfying that $\delta(G[V_i])\geq k_i$ for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity of deciding…

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split…

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $\deg(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in…

We study the problem of partitioning the edge set of the complete graph into bipartite subgraphs under certain constraints defined by forbidden subgraphs. These constraints lead to both classical problems, such as partitioning into…

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erd\H{o}s' Sparse Half Conjecture. Among others we prove the following variant of it: For every sufficiently large even integer $n$ the…

Given a graph G, a subset M of V (G) is a module of G if for each v \in V (G) \diagdownM, v is adjacent to all the elements of M or to none of them. For instance, V(G), \varnothing and {v} (v \in V(G)) are modules of G called trivial. Given…

The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown…

A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is…

An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so…

An internal partition of an $n$-vertex graph $G=(V,E)$ is a partition of $V$ such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every $d$-regular graph with $n>N(d)$…

This work examines the problem of clique enumeration on a graph by exploiting its clique covers. The principle of inclusion/exclusion is applied to determine the number of cliques of size $r$ in the graph union of a set $\mathcal{C} =…

The general position problem in graph theory asks for the largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. In this paper we consider a variant of the general position…

A dominating induced matching, also called an efficient edge domination, of a graph $G=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges is a subset $F \subseteq E$ of edges in the graph such that no two edges in $F$ share a common endpoint…

In this paper, we study commutative zero-divisor semigroups determined by graphs. We prove a uniqueness theorem for a class of graphs. We show two classes of graphs that have no corresponding semigroups. In particular, any complete graph…

Let $G = (V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say that $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A…

Many load balancing problems that arise in scientific computing applications ask to partition a graph with weights on the vertices and costs on the edges into a given number of almost equally-weighted parts such that the maximum boundary…