Related papers: Degree-constrained 2-partitions of graphs

Given $k\ge 1$, a $k$-proper partition of a graph $G$ is a partition ${\mathcal P}$ of $V(G)$ such that each part $P$ of ${\mathcal P}$ induces a $k$-connected subgraph of $G$. We prove that if $G$ is a graph of order $n$ such that…

Let $G = (V, E)$ be a connected graph with maximum degree $k\geq 3$ distinct from $K_{k+1}$. Given integers $s \geq 2$ and $p_1,\ldots,p_s\geq 0$, $G$ is said to be $(p_1, \dots, p_s)$-partitionable if there exists a partition of $V$ into…

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at…

Given a graph $G$ and a non trivial partition $(V_1,V_2)$ of its vertex-set, the satisfaction of a vertex $v\in V_i$ is the ratio between the size of it's closed neighborhood in $V_i$ and the size of its closed neighborhood in $G$. The…

Given nonnegative integers, $s$ and $k$, an $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of $V_G$ such that $G[A]$ and $\overline{G[B]}$ are complete multipartite graphs with at most $s$ and $k$ parts, respectively. If $s$…

We prove that if $G$ is a graph and $r_1, ..., r_k \in \mathbb{Z}_{\geq 0}$ such that $\sum_{i=1}^k r_i \geq \Delta(G) + 2 - k$ then $V(G)$ can be partitioned into sets $V_1, ..., V_k$ such that $\Delta(G[V_i]) \leq r_i$ and $G[V_i]$…

We prove that if $G$ is a vertex critical graph with $\chi(G) \geq \Delta(G) + 1 - p \geq 4$ for some $p \in \mathbb{N}$ and $\omega(\fancy{H}(G)) \leq \frac{\chi(G) + 1}{p + 1} - 2$, then $G = K_{\chi(G)}$ or $G = O_5$. Here $\fancy{H}(G)$…

We define a $(V_1, V_2, \ldots, V_k)$-partition for a given graph $H$ and graphical properties $P_1, P_2, \ldots, P_k$ as a partition where each $V_i$ induces a subgraph of $H$ with property $P_i$. Matamala (2007) extended this result by…

In 1996, Michael Stiebitz proved that if $G$ is a simple graph with $\delta(G)\geq s+t+1$ and $s,t\in \mathbb{Z}_{\geq 0}$, then $V(G)$ can be partitioned into two sets $A$ and $B$ such that $\delta(G[A])\geq s$ and $\delta(G[B])\geq t$. In…

We prove that there exists an absolute constant $C>0$ such that, for any positive integer $k$, every graph $G$ with minimum degree at least $Ck$ admits a vertex-partition $V(G)=S\cup T$, where both $G[S]$ and $G[T]$ have minimum degree at…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

In this paper, we investigate bounds for the following judicious $k$-partitioning problem: Given an edge-weighted graph $G$, find a $k$-partition $(V_1,V_2,\dots ,V_k)$ of $V(G)$ such that the total weight of edges in the heaviest induced…

A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…

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 $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex…

Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the…

An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\ldots,G[V_k])$ of $G$ such that $(V_1,V_2,\ldots,V_k)$ is a partition of $V(G)$ and $-1\le |V_i|-|V_j|\le 1$ for all $1\le i<j\le k$. We prove…

The $k$-deck of a graph is its multiset of induced subgraphs on $k$ vertices. We prove that $n$-vertex graphs with maximum degree $2$ have the same $k$-decks if each cycle has at least $k+1$ vertices, each path component has at least $k-1$…

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 -…

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number…

Let $G$ be a graph of order $n$. The maximum and minimum degree of $G$ are denoted by $\Delta$ and $\delta$ respectively. The \emph{path partition number} $\mu (G)$ of a graph $G$ is the minimum number of paths needed to partition the…