Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs

A Survey on Monochromatic Connections of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it. In this survey, we attempt to bring together all the results that dealt with it. We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.Comment: 26 pages, 3 figure

NP-completeness results for partitioning a graph into total dominating sets

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by d(t) (G). We extend considerably the known hardness results by showing it is NP-complete to decide whether d(t) (G) >= 3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k >= 3, it is NP-complete to decide whether d(t) (G) >= k, where G is split or k-regular. In particular, these results complement recent combinatorial results regarding d(t) (G) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2(n)n(O(1)) time, and derive even faster algorithms for special graph classes. (C) 2018 Elsevier B.V. All rights reserved.Peer reviewe

Total kk-Rainbow domination numbers in graphs

Let $k\geq 1$ be an integer‎, ‎and let $G$ be a graph‎. ‎A {\it‎ ‎$k$-rainbow dominating function} (or a {\it $k$-RDF}) of $G$ is a‎ ‎function $f$ from the vertex set $V(G)$ to the family of all subsets‎ ‎of $\{1,2,\ldots‎ ,‎k\}$ such that for every $v\in V(G)$ with‎ ‎$f(v)=\emptyset$‎, ‎the condition $\bigcup_{u\in‎ ‎N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled‎, ‎where $N_{G}(v)$ is‎ ‎the open neighborhood of $v$‎. ‎The {\it weight} of a $k$-RDF $f$ of‎ ‎$G$ is the value $\omega (f)=\sum _{v\in V(G)}|f(v)|$‎. ‎A $k$-rainbow‎ ‎dominating function $f$ in a graph with no isolated vertex is called‎ ‎a {\em total $k$-rainbow dominating function} if the subgraph of $G$‎ ‎induced by the set $\{v \in V(G) \mid f (v) \not =\emptyset\}$ has no isolated‎ ‎vertices‎. ‎The {\em total $k$-rainbow domination number} of $G$‎, ‎denoted by‎ ‎$\gamma_{trk}(G)$‎, ‎is the minimum weight of a total $k$-rainbow‎ ‎dominating function on $G$‎. ‎The total $1$-rainbow domination is the‎ ‎same as the total domination‎. ‎In this paper we initiate the‎ ‎study of total $k$-rainbow domination number and we investigate its‎ ‎basic properties‎. ‎In particular‎, ‎we present some sharp bounds on the‎ ‎total $k$-rainbow domination number and we determine the total‎ ‎$k$-rainbow domination number of some classes of graphs‎.

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs