Related papers: Dominating Hadwiger's Conjecture for graphs $G$ wi…

Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $\delta(G)$ and chromatic index $\chi'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $\chi'(G)=\D(G)+1$ and $\chi'(H)\textless \chi'(G)$ for…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number…

We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields…

We prove that the complement of any non-separating planar graph of order $2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest possible with this property. To illustrate the necessity of the non-separating hypothesis, we…

For a real number $c > 4$, we prove that every graph $G$ with $\alpha(G) \leq 2$ and $|V(G)| \geq ct$ has a matching $M$ with $|M| = t$ such that the number of non-adjacent pairs of edges in $M$ is at most: \begin{equation*} \left(…

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…

For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the…

In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the…

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must…

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number…

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a…

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…

For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{dominates} $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written…

In this paper we define and study a new family of graphs that generalises the notions of line graphs and path graphs. Let $G$ be a graph with no loops but possibly with parallel edges. An \emph{$\ell$-link} of $G$ is a walk of $G$ of length…

Let $G=(V,E)$ be a simple undirected graph. $G$ is a circulant graph defined on $V=\mathbb{Z}_n$ with difference set $D\subseteq \{1,2,\ldots,\lfloor\frac{n}{2}\rfloor\}$ provided two vertices $i$ and $j$ in $\mathbb{Z}_n$ are adjacent if…

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$…

Let $\gamma(G)$ denote the domination number of graph $G$. Let $G$ and $H$ be graphs and $G\Box H$ their Cartesian product. For $h\in V(H)$ define $G_h=\{(g,h)\,|\,g\in V(G)\}$ and call this set a $G$-layer of $G\Box H$. We prove the…

A domination coloring of a graph $G$ is a proper vertex coloring of $G$ such that each vertex of $G$ dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all…

A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted…

Given graphs $G, H_1, H_2$, we write $G \rightarrow ({H}_1, H_2)$ if every $\{$red, blue$\}$-coloring of the edges of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G…