Related papers: Vizing's conjecture: a two-thirds bound for claw-f…

We prove that $\chi(G) \le \lceil (\Delta+1)/2\rceil+1$ for any triangle-free graph $G$ of maximum degree $\Delta$ provided $\Delta \ge 524$. This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a…

In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to…

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our…

Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such…

For a graph $G = (V, E)$, the $\gamma$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent if they differ by a single vertex and the two…

A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all…

A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $\gamma(G)$, is the minimum order of a dominating set. A subset $R$…

Let $G$ be a graph with degree sequence $d_1\geq \ldots \geq d_n$. Slater proposed $s\ell(G)=\min\{ s: (d_1+1)+\cdots+(d_s+1)\geq n\}$ as a lower bound on the domination number $\gamma(G)$ of $G$. We show that deciding the equality of…

A subset $D$ of vertices of a graph $G$ is a \textit{dominating set} if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The \textit{dominating number}, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating…

Let $G$ be an undirected graph. An edge of $G$ dominates itself and all edges adjacent to it. A subset $E'$ of edges of $G$ is an edge dominating set of $G$, if every edge of the graph is dominated by some edge of $E'$. We say that $E'$ is…

A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ is a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free graph has a 2-factor. The toughness condition of this result is…

A dominating set $D$ in a digraph is a set of vertices such that every vertex is either in $D$ or has an in-neighbour in $D$. A dominating set $D$ of a digraph is locating-dominating if every vertex not in $D$ has a unique set of…

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with…

A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of…

A set $D$ of vertices in a graph $G$ is called dominating if every vertex of $G$ is either in $D$ or adjacent to a vertex of $D$. The paired domination number $\gamma_{\mathrm{pr}}(G)$ of $G$ is the minimum size of a dominating set whose…

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which…

Let $\gamma_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $\gamma_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if…

We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).

Since Reed conjectured in 1996 that the domination number of a connected cubic graph of order $n$ is at most $\lceil \frac13 n \rceil$, the domination number of cubic graphs has been extensively studied. It is now known that the conjecture…

Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by {\gamma}pr(G), is the size of…