The upper domination Ramsey number u(m,n) is the smallest integer p such that every 2-coloring of the edges of Kp with color red and blue, Γ(B) ≥ m or Γ(R) ≥ n, where B and R is the subgraph of Kp induced by blue and red edges, respectively; Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. In this paper, we show that u(4, 4) ≤ 15.

A subset Q ⊆ V (G) is a dominating set of a graph G if each vertex in V (G) is either in Q or is adjacent to a vertex in Q. A dominating set Q of G is minimal if Q contains no dominating set of G as a proper subset. In this paper we study the number of minimal dominating sets in some classes of trees. Mathematics Subject Classification: 05C69

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V,E) with vertex weights w such that ∑k i=1 maxv∈Ci w(vi) is minimized, where C1, . . . , Ck are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be...

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i < j , every vertex ofG colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by (G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining (G) k, for any fixed in...

In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color clas...

Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex c...

Given an undirected graph G = (V, E), the Graph Coloring Problem (GCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. A subset of V is called a stable set if no two adjacent vertices belong to it. A k coloring of G is a coloring which uses k colors, and corresponds to a partition of V into k stab...

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...

In this paper, the concept of incidence domination number of graphs  is introduced and the incidence dominating set and  the incidence domination number  of some particular graphs such as  paths, cycles, wheels, complete graphs and stars are studied.

An acyclic edge-coloring of a graph G is a proper edge-coloring of G such that the subgraph induced by any two color classes is acyclic. The acyclic chromatic index, χa(G), is the smallest number of colors allowing an acyclic edge-coloring of G. Clearly χa(G) ≥ ∆(G) for every graph G. Cohen, Havet, and Müller conjectured that there exists a constant M such that every planar graph with ∆(G) ≥M h...