Related papers: Total Forcing and Zero Forcing in Claw-Free Cubic …

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours…

Given a graph $G$ and a real number $0\le p\le 1$, we define the random set $B_p(G)\subset V(G)$ by including each vertex independently and with probability $p$. We investigate the probability that the random set $B_p(G)$ is a zero forcing…

Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$,…

Let $2\le k\in\mathbb{Z}$. A total coloring of a$k$-regular simple graph via $k+1$ colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction…

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…

Let $G$ be a graph. A total dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $G$ is adjacent to a vertex in $S$. Recently, the following question was proposed: "Is it true that every connected cubic…

A path in an edge-colored graph is called \emph{conflict-free} if it contains at least one color used on exactly one of its edges. An edge-colored graph $G$ is \emph{conflict-free connected} if for any two distinct vertices of $G$, there is…

An injective $k$-edge-coloring of a graph $G$ is a mapping $\phi$: $E(G)\rightarrow\{1,2,...,k\}$, such that $\phi(e)\ne\phi(e')$ if edges $e$ and $e'$ are at distance two, or are in a triangle. The smallest integer $k$ such that $G$ has an…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

The anti-forcing number of a perfect matching $M$ of a graph $G$ is the minimal number of edges not in $M$ whose removal to make $M$ as a unique perfect matching of the resulting graph. The set of anti-forcing numbers of all perfect…

In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the…

Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul\'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges…

Zero forcing is a graph coloring process that is used to model spreading phenomena in real-world scenarios. It can also be viewed as a single-player combinatorial game on a graph, where the player's goal is to select a subset of vertices 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…

Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free…

We prove that every claw-free graph $G$ that doesn't contain a clique on $\Delta(G) \geq 9$ vertices can be $\Delta(G) - 1$ colored.

Given two graphs $F$ and $G$, an $F$-WORM coloring of $G$ is an assignment of colors to its vertices in such a way that no $F$-subgraph of $G$ is monochromatic or rainbow. If $G$ has at least one such coloring, then it is called $F$-WORM…

Probabilistic zero forcing is a graph coloring process in which blue vertices "infect" (color blue) white vertices with a probability proportional to the number of neighboring blue vertices. We introduce reversion probabilistic zero forcing…

An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) denote the number of vertices of G. A graph is called claw-free if it does not have a subgraph isomorphic to the graph with 4 vertices…

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its…