A star k-coloring of a graph G is proper (vertex) such that the vertices on path length three receive at least colors. Given G, its chromatic number, denoted χs(G), minimum integer k for which admits k-coloring. Studying coloring sparse graphs an active area research, especially in terms maximum average degree graph; degree, mad(G), max2|E(H)||V(H)|:H⊂G. It known if mad(G)<83, then χs(G)≤6 (Kün...

In FOCS 2002, Even et al. showed that any set of n discs in the plane can be Conflict-Free colored with a total of at most O(log n) colors. That is, it can be colored with O(log n) colors such that for any (covered) point p there is some disc whose color is distinct from all other colors of discs containing p. They also showed that this bound is asymptotically tight. In this paper we prove the ...

In a proper vertex coloring of a graph a subgraph is colorful if its vertices are colored with different colors. It is well-known that in every proper coloring of a k-chromatic graph there is a colorful path Pk on k vertices. If the graph is k-chromatic and triangle-free then in any proper coloring there is also a path Pk which is an induced subgraph. N.R. Aravind conjectured that these results...

For integers k and l, each greater than 1, suppose that p is a prime with p ≡ 1 (mod k) and that the kth-power classes mod p induce a coloring of the integer segment [0, p− 1] that admits no monochromatic occurrence of l consecutive members of an arithmetic progression. Such a coloring can lead to a coloring of [0, (l − 1)p] that is similarly free of monochromatic l-progressions, and, hence, ca...

Let r,k?1 be two integers. An r-hued k-coloring of the vertices a graph G=(V,E) is proper vertices, such that, for every vertex v?V, number colors in its neighborhood at least min{dG(v),r}, where dG(v) degree v. We prove existence an (r+1)-coloring planar graphs with girth 8 r?9. As corollary, maximum ??9 and admits 2-distance (?+1)-coloring.

We combine the concepts of list colorings of graphs with the concept of defective colorings of graphs and introduce the concept of defective list colorings. We apply these concepts to vertex colorings of various classes of planar graphs. A defective coloring with defect d is a coloring of the vertices such that each color class corresponds to an induced subgraph with maximum degree at most d. A...

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a ke...

For a given graph G, the square of G, denoted by G2, is a graph with the vertex set V(G) such that two vertices are adjacent if and only if the distance of these vertices in G is at most two. A graph G is called squared if there exists some graph H such that G= H2. A function f:V(G) {0,1,2…, k} is called a coloring of G if for every pair of vertices x,yV(G) with d(x,y)=1 we have |f(x)-f(y)|2 an...

For an integer r > 0, a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex of degree at least r in G will be adjacent to vertices with at least r different colors. The smallest integer k for which a graph G has a conditional k-coloring is the r-conditional chromatic number χr(G). In this paper, the behavior and bounds of conditional chrom...

A coloring of a graph G is an assignment of colors to the vertices of G such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring of G is a coloring such that no cycle of G receives exactly two colors, and the acyclic chromatic number χA(G) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determini...