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The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G.… (More)

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic k-coloring can be refined to a star coloring with at most (2k 2 − k) colors.… (More)

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v 1 , v 2 ,. .. , v k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v 1 , v 2 ,. .. , v k in this order. Define f (k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph.… (More)

A proper vertex coloring of a graph is equitable if the sizes of its color classes differ by at most one. In this paper, we prove that if G is a graph such that for each edge xy ∈ E(G), the sum d(x) + d(y) of the degrees of its ends is at most 2r + 1, then G has an equitable coloring with r + 1 colors. This extends the Hajnal–Szemerédi Theorem on graphs… (More)

1. INTRODUCTION. Suppose that Alice wants to color a planar map using four colors in a proper way, that is, so that any two adjacent regions get different colors. Despite the fact that she knows for certain that it is eventually possible, she may fail in her first attempts. Indeed, there are usually many proper partial colorings not extend-able to proper… (More)

An explicit definition of a 1-factorization of B k (the bipartite graph defined by the k-and (k+l)-element subsets of [2k+l]), whose constituent matchings are defined using addition modulo k + 1, is introduced. We show that the matchings are invariant under rotation (mapping under ~r = (1, 2, 3 ..... 2k + 1)), describe the effect of reflection (mapping… (More)