Edge coloring
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A 3-edge-coloring of the Desargues graph.
In graph theory, an edge coloring of a graph is an assignment of “colors” to the edges of the graph so that no two adjacent edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.
By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree Δ or Δ+1. For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always Δ, and for multigraphs, the number of colors may be as large as 3Δ/2. There are polynomial time algorithms that construct optimal colorings of bipartite graphs, and colorings of non-bipartite simple graphs that use at most Δ+1 colors; however, the general problem of finding an optimal edge coloring is NP-complete and the fastest known algorithms for it take exponential time. Many variations of the edge coloring problem, in which an assignments of colors to edges must satisfy other conditions than non-adjacency, have been studied. Edge colorings have applications in scheduling problems and in frequency assignment for fiber optic networks.
By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree Δ or Δ+1. For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always Δ, and for multigraphs, the number of colors may be as large as 3Δ/2. There are polynomial time algorithms that construct optimal colorings of bipartite graphs, and colorings of non-bipartite simple graphs that use at most Δ+1 colors; however, the general problem of finding an optimal edge coloring is NP-complete and the fastest known algorithms for it take exponential time. Many variations of the edge coloring problem, in which an assignments of colors to edges must satisfy other conditions than non-adjacency, have been studied. Edge colorings have applications in scheduling problems and in frequency assignment for fiber optic networks. |
[edit] Examples
A cycle graph may have its edges colored with two colors if the length of the cycle is even: simply alternate the two colors around the cycle. However, if the length is odd, three colors are needed.[1]
Geometric construction of a 7-edge-coloring of the complete graph K8. Each of the seven color classes has one edge from the center to a polygon vertex, and three edges perpendicular to it.
A complete graph Kn with n vertices may have its edges colored with n − 1 colors when n is an even number; this is a special case of Baranyai's theorem. Soifer (2008) provides the following geometric construction of a coloring in this case: place n points at the vertices and center of a regular (n − 1)-sided polygon. For each color class, include one edge from the center to one of the polygon vertices, and all of the perpendicular edges connecting pairs of polygon vertices. However, when n is odd, n colors are needed: each color can only be used for (n − 1)/2 edges, a 1/n fraction of the total.[2]
Several authors have studied edge colorings of the odd graphs, n-regular graphs in which the vertices represent teams of n − 1 players selected from a pool of 2n - 1 players, and in which the edges represent possible pairings of these teams (with one player left as "odd man out" to referee the game). The case that n = 3 gives the well-known Petersen graph. As Biggs (1972) explains the problem (for n = 6), the players wish to find a schedule for these pairings such that each team plays each of its six games on different days of the week, with Sundays off for all teams; that is, formalizing the problem mathematically, they wish to find a 6-edge-coloring of the 6-regular odd graph O6. When n is 3, 4, or 8, an edge coloring of On requires n + 1 colors, but when it is 5, 6, or 7, only n colors are needed.[3]

Definitions
As with its vertex counterpart, an edge coloring of a graph, when mentioned without any qualification, is always assumed to be a proper coloring of the edges, meaning no two adjacent edges are assigned the same color. Here, two edges are considered to be adjacent when they share a common vertex. An edge coloring of a graph G may also be thought of as equivalent to a vertex coloring of the line graph L(G), the graph that has a vertex for every edge of G and an edge for every pair of adjacent edges in G.
A proper edge coloring with k different colors is called a (proper) k-edge-coloring. A graph that can be assigned a (proper) k-edge-coloring is said to be k-edge-colorable. The smallest number of colors needed in a (proper) edge coloring of a graph G is the chromatic index, or edge chromatic number, χ′(G). The chromatic index is also sometimes written using the notation χ1(G); in this notation, the subscript one indicates that edges are one-dimensional objects. A graph is k-edge-chromatic if its chromatic index is exactly k. The chromatic index should not be confused with the chromatic number χ(G) or χ0(G), the minimum number of colors needed in a proper vertex coloring of G.
Unless stated otherwise all graphs are assumed to be simple, in contrast to multigraphs in which two or more edges may connecting the same pair of endpoints and in which there may be self-loops. For many problems in edge coloring, simple graphs behave differently from multigraphs, and additional care is needed to extend theorems about edge colorings of simple graphs to the multigraph case
The direct sum of graph algebras
From CanisiusmathWiki
Let E,F be row-finite directed graphs. Then

Proof:
Recall that the vertex set of the union is the disjoint union of the vertex sets (see A Reference to Some Common Binary Operations on Graphs).
C * (E) is generated by some Cuntz-Krieger E-family {S1,P1} and C * (F) is generated by some Cuntz-Krieger F-family {S2,P2} (Proposition 1.21 in [1]). So and .
By proposition 1.12 in [1], if , then . Hence by proposition A.7 in [1],

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This last equality is seen by noting that a universal Cuntz-Krieger -family will be the union of universal Cuntz-Krieger E and F-families.
Loomis–Whitney inequality
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In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d-dimensional set by the sizes of its (d – 1)-dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas.
The result is named after the American mathematicians L. H. Loomis and Hassler Whitney, and was published in 1949. Contents [hide] * 1 Statement of the inequality * 2 A special case * 3 Generalizations * 4 References |
[edit] Statement of the inequality
Fix a dimension d ≥ 2 and consider the projections
For each 1 ≤ j ≤ d, let
Then the Loomis–Whitney inequality holds:
Equivalently, taking
[edit] A special case
The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space to its "average widths" in the coordinate directions. Let E be some measurable subset of and let be the indicator function of the projection of E onto the jth coordinate hyperplane. It follows that for any point x in E,
Hence, by the Loomis–Whitney inequality,

and hence
The quantity can be thought of as the average width of E in the jth coordinate direction. This interpretation of the Loomis–Whitney inequality also holds if we consider a finite subset of Euclidean space and replace Lebesgue measure by counting measure.
[edit] Generalizations
The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections πj above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension. http://en.wikipedia.org/wiki/Loomis%E2%80%93Whitney_inequality http://davidlivnat.com/wp-content/uploads/2011/01/Theorems-v.11.pdf