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… (More)

A sequence of the form s1s2 . . . sms1s2 . . . sm is called a repetition. A vertex-coloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of… (More)

A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every planar graph of girth 6 (respectively 7,8) can be star… (More)

A multigraph is (k, r)-dense if every k-set spans at most r edges. What is the maximum number of edges exN(n, k, r) in a (k, r)-dense multigraph on n vertices? We determine the maximum possible… (More)

Let n and k be xed positive integers. A collection C of k-sets of n] is a completely separating system if, for all distinct i; j 2 n], there is an S 2 C for which i 2 S and j 6 2 S. Let R(n; k)… (More)

A cut in a graph G is the set of all edges between some set of vertices S and its complement S = V (G) − S. A cut-cover of G is a collection of cuts whose union is E(G) and the total size of a… (More)

A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of… (More)

We determine the maximum number of colors in a coloring of the edges of Km;n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on… (More)