Related papers: On cubic rainbow domination regular graphs
In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of…
Let $G$ be an edge-colored graph on $n$ vertices. For a vertex $v$, the \emph{color degree} of $v$ in $G$, denoted by $d^c(v)$, is the number of colors appearing on the edges incident with $v$. Denote by $\delta^c(G)=\min\{d^c(v):v\in…
Recently the notion of $k$-rainbow total domination was introduced for a graph $G$, motivated by a desire to reduce the problem of computing the total domination number of the generalized prism $G \Box K_k$ to an integer labeling problem on…
From the research of several recent papers, we are concerned with domination number in cubic graphs and give a sufficient condition for Reed's conjecture.
From a research of several recent papers, in the first part, we are concerned with domination number in cubic graphs and give a sufficient condition of Reed's conjecture. In the second part, from a perspective, we study the structure of a…
For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…
In a rainbow version of the classical Tur\'an problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which…
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi that applies to almost optimally bounded colourings. A…
We study the rainbow version of the graph commonness property: a graph $H$ is $r$-rainbow common if the number of rainbow copies of $H$ (where all edges have distinct colors) in an $r$-coloring of edges of $K_n$ is maximized asymptotically…
By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing…
A graph $G$ is called a replication graph of a graph $H$ if $G$ is obtained from $H$ by replacing vertices of $H$ by arbitrary cliques of vertices and then replacing each edge in $H$ by all the edges between corresponding cligues. For a…
Let $G$ be a nontrivial connected and edge-colored graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph $G$ is called rainbow disconnected if for every two distinct…
A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a…
An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are…
A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $ \{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\emptyset $ we have $\bigcup\nolimits_{u\in N(v)}f(u)=\{1,2,...,k\}$. The…
Let $G = (V, E)$ be a graph. A set $S \subseteq V$ is a restrained dominating set (RDS) if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V - S$. The restrained domination number of $G$, denoted by $\gamma_r(G)$,…
For a graph $G$, we define $\sigma_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $\sigma_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct…
An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the \textit{anti-Ramsey number} $ar(n,G)$ is defined to be the minimum number of colors $r$…
Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then…
A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and…