In the work interval colorings [1] of complete bipartite graphs and trees are investigated. The obtained results were announced in [2]. Non defined concepts can be found in [3, 4]. Let G = (V (G), E(G)) be an undirected graph without multiple edges and loops. The degree of a vertex x in G is denoted by dG(x), the greatest degree of vertices – by ∆(G), the chromatic index of G – by χ(G). Interva...

In this paper we introduce a Ramsey type function S(r; a, b, c) as the maximum s such that for any r-coloring of N there is a monochromatic sequence x1, x2, . . . , xs satisfying a homogeneous second order linear recurrence axi + bxi+1 + cxi+2 = 0, 1 6 i 6 s − 2. We investigate S(2; a, b, c) and evaluate its values for a wide class of triples (a, b, c).

Generalizing a result of Erdo s, Gya rfa s and Pyber we show that there exists a constant c such that for any integers r, k 2 and for any coloring of the edges of a complete graph with r colors, its vertices can be partitioned into at most r log r+k) connected monochromatic k-regular subgraphs and vertices. We also show that the same result holds for complete bipartite graphs, generalizing a re...

A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G , denoted by rxk(G). ...

The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...

Let G(V,E) be a k-uniform hypergraph. A hyperedge e ∈ E is said to be properly (r, p) colored by an r-coloring of vertices in V if e contains vertices of at least p distinct colors in the rcoloring. An r-coloring of vertices in V is called a strong (r, p) coloring if every hyperedge e∈E is properly (r, p) colored by the r-coloring. We study the maximum number of hyperedges that can be properly ...

A Grünbaum coloring of a triangulation is an assignment of colors to edges so that the edges about each face are assigned unique colors. In this paper we examine the color induced subgraphs given by a Grünbaum coloring of a triangulation and show that the existence of connected color induced subgraphs is equivalent to the Three Color Theorem. A triangulation of an orientable surface is an embed...

Given a group Γ acting on a set X, a k-coloring φ : X → {1, . . . , k} of X is distinguishing with respect to Γ if the only γ ∈ Γ that fixes φ is the identity action. The distinguishing number of the action Γ, denoted DΓ(X), is then the smallest positive integer k such that there is a distinguishing k-coloring of X with respect to Γ. This notion has been studied in a number of settings, but by ...

We consider a rooted tree R to be a rooted subtree of a given tree T if the tree obtained by replacing the directed arcs of R by undirected edges is a subtree of T . In this work, we study the problem of assigning colors to a given set of rooted subtrees R of a given host tree T such that if any two rooted subtrees share a directed edge, then they are assigned different colors. The objective is...