Colored Trees in Edge-Colored Graphs

The study of problems modeled by edge-colored graphs has given rise to important developments during the last few decades. For instance, the investigation of spanning trees for graphs provide important and interesting results both from a mathematical and an algorithmic point of view (see for instance [1]). From the point of view of applicability, problems arising in molecular biology are often modeled using colored graphs, i.e., graphs with colored edges and/or vertices [6]. Given such an edge-colored graph, original problems translate to extracting subgraphs colored in a specified pattern. The most natural pattern in such a context is that of a proper coloring, i.e., adjacent edges having different colors. Refer to [2,3,5] for a survey of related results and practical applications. Here we deal with some colored versions of spanning trees in edge-colored graphs. In particular, given an edge-colored graph G, we address the question of deciding whether or not it contains properly edge colored spanning trees or rooted edge-colored trees with a given pattern. Formally, let Ic = {1, 2, . . . , c} be a given set of colors, c ≥ 2. Throughout, G denotes an edge-colored simple graph, where each edge is assigned some color i ∈ Ic. The vertex and edge-sets of G are denoted V (G) and E(G), respectively. The order of G is the number n of its vertices. A subgraph of G is said to be properly edge-colored if any two of its adjacent edges differ in color. A tree in G is a subgraph such that its underlying non-colored graph is connected and acyclic. A spanning tree is one covering all vertices of G. From the earlier definitions, a properly edge-colored tree is one such that no two adjacent edges are on a same color. A tree T in G with fixed root r is said to be weakly properly edge-colored if any path in T , from the root r to any leaf is a properly edge-colored one. To facilitate discussions, in the sequel a properly edge-colored (weakly properly edge-colored) tree will be called a strong (weak) tree. Notice that in the case of weak trees, adjacent edges may have the same color, while this may not happen for strong trees. When these trees span the vertex set of G, they are called strong spanning tree (sst) and weak spanning tree (wst).