Related papers: Packing spanning partition-connected subgraphs wit…

Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\pi=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\pi$ is connected for…

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…

The $g$-component edge connectivity $c\lambda_g(G)$ of a non-complete graph $G$ is the minimum number of edges whose deletion results in a graph with at least $g$ components. In this paper, we determine the component edge connectivity of…

Let $G=(V,E)$ be a strongly biconnected directed graph. In this paper we consider the problem of computing an edge subset $H \subseteq E$ of minimum size such that the directed subgraph $(V,H)$ is strongly biconnected.

A subgraph (a spanning subgraph) of a graph G whose all components are 3-vertex paths is called an L-packing (respectively, an L-factor} of G. We discuss the following old PROBLEM (A. Kelmans, 1984). Is the following claim true? (C) If G is…

An {\it additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $…

Given two graphs $G$ and $H$, we define $\textsf{v-cover}_{H}(G)$ (resp. $\textsf{e-cover}_{H}(G)$) as the minimum number of vertices (resp. edges) whose removal from $G$ produces a graph without any minor isomorphic to ${H}$. Also…

Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\in V$ to $r_2\in V$, we define $\partial(P) = r_2-r_1$. If $R, S\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\partial P\otimes \partial…

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors…

Let $G$ be a graph of order $n$. For every $v\in V(G)$, let $E_G(v)$ denote the set of all edges incident with $v$. A signed $k$-submatching of $G$ is a function $f:E(G)\longrightarrow \{-1,1\}$, satisfying $f(E_G(v))\leq 1$ for at least…

An $L(2,1)$-labeling of a graph $G=(V,E)$ is a function $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two, and the labels on vertices at distance two differ by…

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a simple graph, an $L(2,1)$-labeling of $\mathcal{G}$ is an assignment of labels from nonnegative integers to vertices of $\mathcal{G}$ such that adjacent vertices get labels which differ by…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…

Given an undirected graph $G = (V_G, E_G)$ and a fixed "pattern" graph $H = (V_H, E_H)$ with $k$ vertices, we consider the $H$-Transversal and $H$-Packing problems. The former asks to find the smallest $S \subseteq V_G$ such that the…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

We show that for every $n\in\mathbb N$ and $\log n\le d\le n$, if a graph $G$ has $N=\Theta(dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.