Let G be an edge-colored graph with n vertices. A rainbow subgraph is a subgraph whose edges have distinct colors. The rainbow edge-chromatic number of G, written χ̂′(G), is the minimum number of rainbow matchings needed to cover E(G). An edgecolored graph is t-tolerant if it contains no monochromatic star with t+1 edges. If G is t-tolerant, then χ̂′(G) < t(t+ 1)n lnn, and examples exist with χ̂′(...

Let m(r, k) denote the minimum number of edges in an r-uniform hypergraph that is not k-colorable. We give a new lower bound on m(r, k) for fixed k and large r. Namely, we prove that if k 2, then m(r, k) (k)k(r/ln r) . © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 24: 1–10, 2004

Clustering is an important task in the process of data analysis which can be viewed as a data modeling technique that provides an attractive mechanism to automatically find the hidden structure of large data sets (Jain et al., 1999). Informally, this task consists of the division of data items (objects, instances, etc.) into groups or categories, such that all objects in the same group are simi...

The Lights Out game on a graph G is played as follows. Begin with a (not necessarily proper) coloring of V (G) with elements of Z2. When a vertex is toggled, that vertex and all adjacent vertices change their colors from 0 to 1 or vice-versa. The game is won when all vertices have color 0. The winnability of this game is related to the existence of a parity dominating set. We generalize this ga...

Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreover, for every fixed k ≥ 0, such problems can be solved in linear time on graphs of treewidth at ...

An acyclic coloring of a graph is a proper coloring such that any two color classes induce a forest. A star coloring of a graph is an acyclic coloring with the further restriction that the forest induced by any two color classes is a disjoint collection of stars. We consider the behavior of these problems when restricted to certain classes of graphs. In particular, we give characterizations of ...

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in [3]. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of [3] by sh...

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating diffi...

Abstract For a graph G , two dominating sets D and $$D'$$ D ′ in non-negative integer k the set is said to -transform if there sequence $$D_0,\ldots ,D_\ell $$ 0 , … ℓ ...