Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a compe...

A (proper) k-coloring of agraph G is a partition = {V1; V2; : : : ; Vk} of V (G) into k independent sets, called color classes. In a k-coloring , a vertex v∈Vi is called a Grundy vertex if v is adjacent to at least one vertex in color class Vj , for every j, j¡ i. A k-coloring is called a Grundy coloring if every vertex is a Grundy vertex. A k-coloring is called a partial Grundy coloring if eve...

A total dominator coloring of a graph G is proper in which each vertex the adjacent to every some color class. The chromatic number minimum classes coloring. In this paper, we study on middle graphs by giving several bounds for case general and trees. Moreover, calculate explicitly known families graphs.

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

Given a vertex-colored graph, we say path is rainbow vertex if all its internal vertices have distinct colors. The graph vertex-connected there between every pair of vertices. In the Rainbow Vertex Coloring (RVC) problem want to decide whether given can be colored with at most k colors so that becomes vertex-connected. This known NP -complete even in very restricted scenarios, and few efficient...

A hypergraph is a set V of vertices and a set E of non-empty subsets of V , called hyperedges. Unlike graphs, hypergraphs can perform higher-order interactions in social and communication networks. Directed hypergraphs are much like directed graphs. Colors are used to distinguish the classes. Coloring a hypergraph H must assign atleast two different colors to the vertices of every hyperedge. Th...

An acyclic k-coloring of a graph G is a proper vertex coloring of G which uses at most k colors such that the graph induced by the union of every two color classes is a forest. In this paper, we mainly prove that every 5-connected graph with maximum degree five is acyclically 8-colorable, improving partially [5].

An acyclic k-coloring of a graph G is a proper vertex coloring of G, which uses at most k colors, such that the graph induced by the union of every two color classes is a forest. In this note, we prove that every graph with maximum degree six is acyclically 11-colorable, thus improving the main result of [12].

a proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. a graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $g$ such that each vertex receives a color from its own list. in this paper, we prov...

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a compe...