Related papers: Degree-constrained 2-partitions of graphs

For a given graph $H$ and the graphical properties $P_1, P_2,\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\ldots,V_k$, such that for each…

Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$…

A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We…

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…

Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $\delta(H, n)$ be the minimum…

A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a…

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$…

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal…

Let $\mathcal{G}_{\alpha}$ be a hereditary graph class (i.e, every subgraph of $G_{\alpha}\in \mathcal{G}_{\alpha}$ belongs to $\mathcal{G}_{\alpha}$) such that every graph $G_{\alpha}$ in $\mathcal{G}_{\alpha}$ has minimum degree at most…

A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is…

For a given $\pi=(\pi_0, \pi_1,..., \pi_k) \in \{0, 1, *\}^{k+1}$, we want to determine whether an input $k$-uniform hypergraph $G=(V, E)$ has a partition $(V_1, V_2)$ of the vertex set so that for all $X \subseteq V$ of size $k$, $X \in E$…

We give an FPT algorithm for deciding whether the vertex set a digraph $D$ can be partitioned into two disjoint sets $V_1,V_2$ such that the digraph $D[V_1]$ induced by $V_1$ has a vertex that can reach all other vertices by directed paths,…

The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an…

A {\bf $\mathbf{k}$-majority coloring} of a digraph $D=(V,A)$ is a coloring of $V$ with $k$ colors so that each vertex $v\in V$ has at least as many out-neighbours of color different from its own color as it has out-neighbours with the same…

Borodin and Kostochka proved that for $d_2 \geq 2d_1+2$ and a graph $G$ where every subgraph $H$ satisfies $$ e(H) < \left(2 - \frac{d_2+2}{(d_1+2)(d_2+1)}\right)n(H) + \frac{1}{d_2+1} $$ has a vertex partition $V(G) = V_1 \cup V_2$ such…

A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as…

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a special spanning tree in which the distance between any two adjacent vertices…

We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average…

In this note we prove that for every integer $k$, there exist constants $g_{1}(k)$ and $g_{2}(k)$ such that the following holds. If $G$ is a graph on $n$ vertices with maximum degree $\Delta$ then it contains an induced subgraph $H$ on at…

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be…