Given an edge colouring of a graph with a set of m colours, we say that the graph is (exactly) m-coloured if each of the colours is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number m > 2 and for all sufficiently large k, there is a k-colouring of the complete graph on N such that no complete infinite subgraph is exac...

The Ramsey number R(G1; G2; : : : ; Gn) is the smallest integer p such that for any n-edge coloring (E1; E2; : : : ; En) of Kp; Kp[Ei] contains Gi for some i, Gi as a subgraph in Kp[Ei]. Let R(m1; m2; : : : ; mn):=R(Km1 ; Km2 ; : : : ; Kmn); R(m; n):=R(m1; m2; : : : ; mn) if mi=m for i=1; 2; : : : ; n. A formula is obtained for R(G1; G2; : : : ; Gn). c © 2001 Elsevier Science B.V. All rights re...

We explore properties of edge colorings of graphs defined by set intersections. An edge coloring of a graphG with vertex set V ={1,2, . . . ,n} is called transitive if one can associate sets F1,F2, . . . ,Fn to vertices of G so that for any two edges ij,kl∈E(G), the color of ij and kl is the same if and only if Fi∩Fj =Fk∩Fl. The term transitive refers to a natural partial order on the color set...

The topic of this paper is the rectangle-free coloring of grids using four colors which is equivalent to the edge coloring of complete bipartite graphs without complete monochromatic subgraphs K2,2. Despite a strong mathematical background it is not known whether rectangle-free 4-colorable grids exist for five large grid sizes. We present in this paper an approach that solves the most complex p...

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph Kn that has no rainbow spanning subgraph with diameter at most t. We prove s(n, t) = (n−2 2 ) +1 for n, t ≥ 3, while s(n, 2) = (n−2 2 )

The problem of edge coloring a bipartite graph is to color the edges so that adjacent edges receive di erent colors An optimal algorithm uses the minimum number of colors to color the edges We consider several approximation algorithms for edge coloring bipartite graphs and show tight bounds on the number of colors they use in the worst case We also brie y consider the constrained edge coloring ...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a(G) ≤ ∆ + 2, where ∆ = ∆(G) denotes the maximum degree of the gra...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b-vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In thi...