Here we prove a stability version of a Ramsey-type Theorem for paths. Thus in any 2-coloring of the edges of the complete graph Kn we can either find a monochromatic path substantially longer than 2n/3, or the coloring is close to the extremal coloring.

It is proved in this note that the Grundy number, T(G), and the ochromatic number, x’(G), are the same for any graph G. An n-coloring of a graph G = (V, E) is a function f from I/ onto N ={1,2,..., n} such that, whenever vertices u and u are adjacent. then f(u) f f(u). An n-coloring is complete if for every pair i,j of integers, 1 5 i 5 j 5 n, there exist a pair U, u of adjacent vertices such t...

In a proper edge-coloring of cubic graph, an edge e is normal if the set colors used by five edges incident with end has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless graph 5-edge-coloring, is, 5-edge-coloring such all are normal. this paper, we prove result related to conjecture. parameter μ3 measurement for graphs, introduced Steffen in 2015. Our shows G a...

We investigate a coloring problem, called ordered coloring, in grids and some other families of grid-like graphs. Ordered coloring (also known as vertex ranking) has applications, among other areas, in efficient solving of sparse linear systems of equations and scheduling parallel assembly of products. Our main technical results improve upper and lower bounds for the ordered chromatic number of...

A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 73 admits a 2-distance (Δ + 1)-coloring. This result is tight. This improves previous known results of Dolama and Sopena.