Acyclic coloringIn graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic number A(G) of a graph G is the fewest colors needed in any acyclic coloring of G. Acyclic coloring is often associated with graphs embedded on non-plane surfaces. A(G) ≤ 2 if and only if G is acyclic. Bounds on A(G) in terms of Δ(G), the maximum degree of G, include the following: A(G) ≤ 4 if Δ(G) = 3. A(G) ≤ 5 if Δ(G) = 4. A(G) ≤ 7 if Δ(G) = 5. A(G) ≤ 12 if Δ(G) = 6.
Vertex (graph theory)In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.
Brooks' theoremIn graph theory, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs and cycle graphs of odd length, which require Δ + 1 colors. The theorem is named after R. Leonard Brooks, who published a proof of it in 1941. A coloring with the number of colors described by Brooks' theorem is sometimes called a Brooks coloring or a Δ-coloring.
Caterpillar treeIn graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path. Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by Arthur Hobbs. As colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed." The following characterizations all describe the caterpillar trees: They are the trees for which removing the leaves and incident edges produces a path graph.
Steinitz's theoremIn polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.
Congenital red–green color blindnessCongenital red–green color blindness is an inherited condition that is the root cause of the majority of cases of color blindness. It has no significant symptoms aside from its minor to moderate effect on color vision. It is caused by variation in the functionality of the red and/or green opsin proteins, which are the photosensitive pigment in the cone cells of the retina, which mediate color vision. Males are more likely to inherit red–green color blindness than females, because the genes for the relevant opsins are on the X chromosome.
16-cellIn geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid . It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's polytope.
