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...

A graph is H-free if it does not contain an induced subgraph isomorphic to H . We denote by Pk and Ck the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on kcoloring Pt-free graphs, and almost complete the classificatio...

Given a graph G = (V,E) with strictly positive integer weights ωi on the vertices i ∈ V , a k-interval coloring of G is a function I that assigns an interval I(i) ⊆ {1, · · · , k} of ωi consecutive integers (called colors) to each vertex i ∈ V . If two adjacent vertices x and y have common colors, i.e. I(i)∩ I(j) 6= ∅ for an edge [i, j] in G, then the edge [i, j] is said conflicting. A k-interv...

Bipolar fuzzy graph (BFG) coloring techniques are used to solve many complex real world problems. The chromatic number of complement BFG is obtained and compared with the corresponding BFGs. This paper an attempt define in a based on strong edges. complete BF tree obtained.

In this paper we investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color α is legal for a vertex if said vertex has no neighbors colored α. Other variations of the game change this definition of a legal color. For a fixed color ...

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...