Let G be a simple graph with vertex set V(G) and edge set E(G). A subset S of V(G) is called an independent set if no two vertices of S are adjacent in G. The minimum number of independent sets which form a partition of V(G) is called chromatic number of G, denoted by χ(G). A subset S of E(G) is called an edge cover of G if the subgraph induced by S is a spanning subgraph of G. The maximum numb...

The Ramsey number R(s, t) for positive integers s and t is the minimum integer n for which every red-blue coloring of the edges of a complete n-vertex graph induces either a red complete graph of order s or a blue complete graph of order t. This paper proves that R(3, t) is bounded below by (1 − o(1))t2/ log t times a positive constant. Together with the known upper bound of (1 + o(1))t2/ log t...

Suppose G is r-colorable and P ⊆ V (G) is such that the components of G[P ] are far apart. We show that any (r + s)-coloring of G[P ] in which each component is s-colored extends to an (r + s)-coloring of G. If G does not contract to K5 or is planar and s ≥ 2, then any (r + s − 1)-coloring of P in which each component is s-colored extends to an (r + s − 1)-coloring of G. This result uses the Fo...

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular, we show that for a given s with a few small exceptions, there is such a coloring with c colors if and only r ≤ t ≤ cs − r where r = blogc(s − 1)c + 1 if s ≤ c or if s ≥ c + 1 and s ≥ c1+blogc(s−1)c − blogcblogc(s− 1)cc and...

We prove that for every k there is a k -chromatic graph with a k-coloring where the neighbors of each color-class form an independent set. This answers a question raised by N. J. A. Harvey and U. S. R. Murty [4]. In fact we find the smallest graph Gk with the required property for

An r-dynamic coloring of a graph G is proper such that every vertex in V(G) has neighbors at least $\min\{d(v),r\}$ different color classes. The chromatic number denoted as $\chi_r (G)$, the k coloring. In this paper we obtain central graph, middle total line para-line and sub-division comb $P_n\odot K_1$ by $C(P_n\odot K_1), M(P_n\odot T(P_n\odot L(P_n\odot P(P_n\odot K_1)$ $S(P_n\odot respect...

The well-known technique of n-coloring a diagram of an oriented link l is generalized using elements of the circle T for colors. For any positive integer r, the more general notion of a (T, r)-coloring is defined by labeling the arcs of a diagram D with elements of the torus Tr−1. The set of (T, r)-colorings of D is an abelian group, and its quotient by the connected component of the idenitity ...

We show that for any two linear homogeneous equations E0, E1, each with at least three variables and coefficients not all the same sign, any 2-coloring of Z+ admits monochromatic solutions of color 0 to E0 or monochromatic solutions of color 1 to E1. We define the 2-color off-diagonal Rado number RR(E0, E1) to be the smallest N such that [1, N ] must admit such solutions. We determine a lower b...

The lower bound for the classical Ramsey number R(4, 8) is improved from 56 to 58. The author has found a new edge coloring of K57 that has no complete graphs of order 4 in the first color, and no complete graphs of order 8 in the second color. The coloring was found using a SAT solver which is based on MiniSat and customized for solving Ramsey problems. Recently Exoo improved the lower bound f...

Let G be a graph and R ⊆ V (G). A proper edge-coloring of a graph G with colors 1, . . . , t is called an R-sequential t-coloring if the edges incident to each vertex v ∈ R are colored by the colors 1, . . . , dG(v), where dG(v) is the degree of the vertex v in G. In this note, we show that if G is a graph with ∆(G) − δ(G) ≤ 1 and χ′(G) = ∆(G) = r (r ≥ 3), then G has an R-sequential r-coloring ...