Let G = (V,E) be a vertex-colored graph, where C is the set of colors used to color V . The Graph Motif (or GM) problem takes as input G, a multisetM of colors built from C, and asks whether there is a subset S ⊆ V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M . The Colorful Graph Motif (or CGM) problem is the special case of GM in which M is a set, an...

An m-distinct-coloring is a proper vertex-coloring c of a graph G if for each vertex v ∈ V , any color appears in at most one of N0(v), N1(v), . . ., and Nm(v), where Ni (v) is the set of vertices at distance i from v. In this note, we show that if G is C2m+1-free which is assigned an (m + 1)-distinct-coloring c, then α(G)c(G)1/m ≥ ( ∑ v c(v) 1/m ) , where c(G) is the number of colors used in c...

A colored lattice L(c) has a geometrical lattice L. A subgroup lattice L' of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L' in L is given by Delta, the determinant of the matrix (t(jk)) that converts L into L'. This is the order of the factor group {L/L'}, and is also the number n of colors present. The crystal systems-i.e., the com...

Many species of birds have feathers that are brilliantly colored without the use of pigments. In these cases, light of specifi c wavelengths is selectively scattered from nanostructures with variations in index of refraction on length-scales of the order of visible light. [ 1 ] This phenomenon is called structural color. The most striking examples of structural color in nature are iridescent co...

An edge-coloring of a graph G1⁄4 ðV ; EÞ is a function c that assigns an integer c(e) (called color) in f 0;1;2;...g to every edge eAE so that adjacent edges are assigned different colors. An edge-coloring is compact if the colors of the edges incident to every vertex form a set of consecutive integers. The deficiency problem is to determine the minimum number of pendant edges that must be adde...

Let (B,C) be an (edge-)colored bipartite graph with bipartition (X,Y ), i.e., B is assigned a mapping C : E(B) → {1, 2, · · · , r}, the set of colors. A matching of B is called heterochromatic if its any two edges have different colors. Let N (S) denote a maximum color neighborhood of S ⊆ V (B). We show that if |N (S)| ≥ |S| for all S ⊆ X, then B contains a heterochromatic matching with cardina...

A mixed hypergraph consists of two families of subsets of the vertex set: the V-edges and the C-edges. In a suitable coloring of a mixed hypergraph, every C-edge has at least two vertices of the same color, and every V-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are called the upper and lower chromatic numbers, X and X, r...

A mixed hypergraph is a triple H = (X, C,D) where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A k-coloring of H is a mapping c : X → [k] such that each C-edge has two vertices with the same color and each D-edge has two vertices with distinct colors. H = (X, C,D) is called a mixed hypertree if there exists a tree T = (X, E) such that ...

In this chapter we focus on image segmentation techniques for some very special images — textile images. They are generated from the color halftoning technique in textile and printing production lines. In contrast with natural color images, textile images have some very distinctive features: (1) generally there are a few dominant colors in a textile image, whereas there may exist hundreds of si...