Related papers: The $k$-representation number of the random graph

Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v \mapsto A_v$ to the vertices $v\in V$ such that $A_u$ and $A_v$ are disjoint if and only if $uv\in E$. The…

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u \in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is…

A unit cube in $k$ dimensional space (or \emph{$k$-cube} in short) is defined as the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A…

A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $\pi(v)(i) \neq \pi(u)(i)$ for each $i \in…

The clique cover number of a graph G is the minimum number of cliques required to cover the edges of graph G. In this paper we consider the random graph G(n,p), for p constant. We prove that with probability 1-o(1), the clique number of…

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…

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus…

For an integer $k\ge 3$, a $k$-path vertex cover of a graph $G=(V,E)$ is a set $T\subseteq V$ that shares a vertex with every path subgraph of order $k$ in $G$. The minimum cardinality of a $k$-path vertex cover is denoted by $\psi_k(G)$.…

In a graph $G$, a vertex dominates itself and its neighbours. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $D$ dominates every vertex of $G$ at least $k$ times. The minimum cardinality among all $k$-tuple…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

We consider $k$-cop-win graphs in the binomial random graph $G(n,1/2).$ It is known that almost all cop-win graphs contain a universal vertex. We generalize this result and prove that for every $k \in N$, almost all $k$-cop-win graphs…

For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$.…

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex…

Let $k\ge 2$ be fixed integer, $0<c<1$ a constant. Consider a graph $G$ with $n$ vertices and average degree $cn$. We answer a question of Simon Griffiths by showing that $G$ has $k$ vertices such that their neighborhoods together cover at…

Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k \subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of $G$ is called the $k$-nearly edge independence number of $G$, and is denoted by…

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…

A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$.The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…