相关论文: Deploying robots with two sensors in $K_{1,6}$-fre…

The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are…

The domatic number of a graph $G$ is the maximum number of pairwise disjoint dominating sets of $G$. We are interested in the LP-relaxation of this parameter, which is called the fractional domatic number of $G$. We study its extremal value…

We show that if $G$ is a simple triangle-free graph with $n\geq 3$ vertices, without a perfect matching, and having a minimum degree at least $\frac{n-1}{2}$, then $G$ is isomorphic either to $C_5$ or to $K_{\frac{n-1}{2},\frac{n+1}{2}}$.

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size…

A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number,…

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…

A non-zero component graph $G(\mathbb{V})$ associated to a finite vector space $\mathbb{V}$ is a graph whose vertices are non-zero vectors of $\mathbb{V}$ and two vertices are adjacent, if their corresponding vectors have at least one…

A 2-factor of a graph is a 2-regular spanning subgraph. For a graph $G$ and an independent set $I$ of $G$, let $\delta_G(I)$ denote the minimum degree of vertices contained in $I$. We show that (1) if every independent set $I$ of $G$…

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…

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of…

A graph $G$ is $\{F_{1}, F_{2},\dots,F_{k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…

The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices…

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…

Let $G$ be a graph. We say that $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. We use $P_t$ and $C_t$ to denote a path…

A vertex partition in which every part induces a 2-connected subgraph is called a 2-proper partition. This concept was introduced by Ferrara et al. in 2013, and Borozan et al. gave the best possible minimum degree condition for the…

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…

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to…

A claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph, and a 2-factor is a 2-regular spanning subgraph of a graph. In 1997, Ryj\'{a}\v{c}ek introduced the closure concept of claw-free graphs, and Hamilton…