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By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree Δ or Δ+1.. There are polynomial time algorithms that construct optimal 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/2Δ/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

[hide]

 1 Examples

 2 Definitions

 3 Relation to matching

 4 Relation to degree

o 4.1 Vizing's theorem

o 4.2 Regular graphs

o 4.3 Planar graphs

o 4.4 Graphs on nonplanar surfaces

o 4.5 Multigraphs

 5 Algorithms

o 5.1 Optimally coloring special classes of graphs

o 5.2 Algorithms that use more than the optimal number of colors

o 5.3 Exact algorithms

 6 Additional properties

 7 Other types of edge coloring

 8 Applications

 9 Open problems

 10 Notes

 11 References

 12 External links

[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 K 8 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 = 3Δ/2 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 3Δ/2, 4, or 8, an edge coloring of O n requires n + 1 colors, but when it is 5, 6, or

7, only n colors are needed.[3Δ/2]

[edit] 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

[edit] Relation to matching

This 3-regular planar graph has 16 vertices and 24 edges, but only 7 edges in any maximum matching Therefore, it requires four colors in any edge coloring.

A matching in a graph G is a set of edges, no two of which are adjacent; a perfect matching is a matching that includes edges touching all of the vertices of the graph, and a maximum matching

is a matching that includes as many edges as possible In an edge coloring, the set of edges with any one color must all be non-adjacent to each other, so they form a matching That is, a proper edge coloring is the same thing as a partition of the graph into disjoint matchings

If the size of a maximum matching in a given graph is small, then many matchings will be needed in order to cover all of the edges of the graph Expressed more formally, this reasoning

implies that if a graph has m edges in total, and if at most β edges may belong to a maximum matching, then every edge coloring of the graph must use at least m/β different colors.[4] For

instance, the 16-vertex planar graph shown in the illustration has m = 24 edges In this graph,

there can be no perfect matching; for, if the center vertex is matched, the remaining unmatched vertices may be grouped into three different connected components with four, five, and five vertices, and the components with an odd number of vertices cannot be perfectly matched However, the graph has maximum matchings with seven edges, so β = 7 Therefore, the number

of colors needed to edge-color the graph is at least 24/7, and since the number of colors must be

an integer it is at least four

For a regular graph of degree k that does not have a perfect matching, this lower bound can be

used to show that at least k + 1 colors are needed.[4] In particular, this is true for a regular graph with an odd number of vertices (such as the odd complete graphs); for such graphs, by the handshaking lemma, k must itself be even However, the inequality χ′ ≥ m/β does not fully explain the chromatic index of every regular graph, because there are regular graphs that do have

perfect matchings but that are not k-edge-colorable For instance, the Petersen graph is regular, with m = 15 and with β = 5 edges in its perfect matchings, but it does not have a 3Δ/2-edge-coloring.

[edit] Relation to degree

[ edit ] Vizing's theorem

The edge chromatic number of a graph G is very closely related to the maximum degree Δ(G), the largest number of edges incident to any single vertex of G Clearly, χ′(G) ≥ Δ(G), for if Δ different edges all meet at the same vertex v, then all of these edges need to be assigned different

colors from each other, and that can only be possible if there are at least Δ colors available to be assigned Vizing's theorem (named for Vadim G Vizing who published it in 1964) states that

this bound is almost tight: for any graph, the edge chromatic number is either Δ(G) or Δ(G) + 1 When χ′(G) = Δ(G), G is said to be of class 1; otherwise, it is said to be of class 2.

For instance, when Δ(G) = 1, the graph G must itself be a matching, with no two edges adjacent, and its edge chromatic number is one That is, all graphs with Δ(G) = 1 are of class 1 When Δ(G) = 2, the graph G must be a disjoint union of paths and cycles; in this case, it can be 2-edge-colored if and only if all of the cycles have even length That is, a graph with Δ(G) = 2 is of class

1 if and only if it is bipartite More generally, according to a theorem of Kőnig (1916), every bipartite graph is of class 1, regardless of its maximum degree However, for non-bipartite graphs with larger maximum degree than two, it is much more difficult to distinguish class 1 graphs from class 2 graphs: Holyer (1981) proved that it is NP-complete to determine whether a graph is of class 1

Several authors have provided additional conditions that classify some graphs as being of class 1

or class 2, but do not provide a complete classification For instance, if the vertices of the

maximum degree Δ in a graph G form an independent set, or more generally if the induced

subgraph for this set of vertices is a forest, then G must be of class 1.[5]

Erdős & Wilson (1977) showed that almost all graphs are of class 1 That is, in the Erdős–Rényi model of random graphs, in which all n-vertex graphs are equally likely, let p(n) be the

probability that an n-vertex graph drawn from this distribution is of class 1; then p(n) approaches one in the limit as n goes to infinity For more precise bounds on the rate at which p(n) converges

to one, see Frieze et al (1988)

A 1-factorization of a k-regular graph, a partition of the edges of the graph into perfect

matchings, is the same thing as a k-edge-coloring of the graph That is, a regular graph has a 1-factorization if and only if it is of class 1 As a special case of this, a 3Δ/2-edge-coloring of a cubic

(3Δ/2-regular) graph is sometimes called a Tait coloring.

Not every regular graph has a 1-factorization; for instance, the Petersen graph does not More generally the snarks are defined as the graphs that, like the Petersen graph, are bridgeless, 3Δ/2-regular, and of class 2

According to the theorem of Kőnig (1916), every bipartite regular graph has a 1-factorization The theorem was stated earlier in terms of projective configurations and was proven by Ernst Steinitz in his PhD thesis

[ edit ] Planar graphs

Vizing (1965) showed that a planar graph is of class 1 if its maximum degree is at least eight In contrast, he observed that for any maximum degree in the range from two to five, there exist planar graphs of class 2 For degree two, any odd cycle is such a graph, and for degree three, four, and five, these graphs can be constructed from platonic solids by replacing a single edge by

a path of two adjacent edges

In Vizing's planar graph conjecture, Vizing (1965) states that all simple, planar graphs with

maximum degree six or seven are of class 1, closing the remaining possible cases Sanders & Zhao (2001) partially proved Vizing's planar graph conjecture by showing that all planar graphs with maximum degree seven are of class 1 Thus, the only case of the conjecture that remains unsolved is that of maximum degree six This conjecture has implications for the total coloring conjecture

The planar graphs of class 2 constructed by subdivision of the platonic solids are not regular: they have vertices of degree two as well as vertices of higher degree The four color theorem, on vertex coloring of planar graphs, is equivalent to the statement that every bridgeless 3Δ/2-regular planar graph is of class one (Tait 1880) This statement is now known to be true, due to the proof

of the four color theorem by Appel & Haken (1976)

In 1969, Branko Grünbaum conjectured that every 3Δ/2-regular graph with a polyhedral embedding

on any two-dimensional oriented manifold such as a torus must be of class one In this context, a polyhedral embedding is a graph embedding such that every face of the embedding is

topologically a disk and such that the dual graph of the embedding is simple, with no self-loops

or multiple adjacencies If true, this would be a generalization of the four color theorem, which was shown by Tait to be equivalent to the statement that 3Δ/2-regular graphs with a polyhedral embedding on a sphere are of class one However, Kochol (2009) showed the conjecture to be false by finding snarks that have polyhedral embeddings on high-genus orientable surfaces Based on this construction, he also showed that it is NP-complete to tell whether a polyhedrally embedded graph is of class 1.[6]

A Shannon multigraph with degree six, edge multiplicity three, and requires nine colors in any edge coloring

For multigraphs, in which multiple parallel edges may connect the same two vertices, results that are similar to but weaker than Vizing's theorem are known relating the edge chromatic number χ′

(G), the maximum degree Δ(G), and the multiplicity μ(G), the maximum number of edges in any

bundle of parallel edges As a simple example showing that Vizing's theorem does not generalize

to multigraphs, consider a Shannon multigraph, a multigraph with three vertices and three

bundles of μ(G) parallel edges connecting each of the three pairs of vertices In this example, Δ(G) = 2μ(G) (each vertex is incident to only two out of the three bundles of μ(G) parallel edges) but the edge chromatic number is 3Δ/2μ(G) (there are 3Δ/2μ(G) edges in total, and every two edges are

adjacent, so all edges must be assigned different colors to each other) In a result that inspired Vizing,[7] Shannon (1949) showed that this is the worst case: χ′(G) ≤ (3Δ/2/2)Δ(G) for any

multigraph G Additionally, for any multigraph G, χ′(G) ≤ Δ(G) + μ(G), an inequality that reduces to Vizing's theorem in the case of simple graphs (for which μ(G) = 1).

[edit] Algorithms

Because the problem of testing whether a graph is class 1 is NP-complete, there is no known polynomial time algorithm for edge-coloring every graph with an optimal number of colors Nevertheless a number of algorithms have been developed that relax one or more of these

criteria: they only work on a subset of graphs, or they do not always use an optimal number of colors, or they do not always run in polynomial time

In the case of bipartite graphs or multigraphs with maximum degree Δ, the optimal number of colors is exactly Δ Cole, Ost & Schirra (2001) showed that an optimal edge coloring of these

graphs can be found in the near-linear time bound O(m log Δ), where m is the number of edges in

the graph; simpler, but somewhat slower, algorithms are described by Cole & Hopcroft (2002) and Alon (2004) The algorithm of Alon (2004) begins by making the input graph regular,

without increasing its degree or significantly increasing its size, by merging pairs of vertices that belong to the same side of the bipartition and then adding a small number of additional vertices and edges Then, if the degree is odd, Alon finds a single perfect matching in near-linear time, assigns it a color, and removes it from the graph, causing the degree to become even Finally, Alon applies an observation of Gabow (1976), that selecting alternating subsets of edges in an Euler tour of the graph partitions it into two regular subgraphs, to split the edge coloring problem into two smaller subproblems, and his algorithm solves the two subproblems recursively The

total time for his algorithm is O(m log m).

For planar graphs with maximum degree Δ ≥ 7, the optimal number of colors is again exactly Δ With the stronger assumption that Δ ≥ 9, it is possible to find an optimal edge coloring in linear time (Cole & Kowalik 2008)

Misra & Gries (1992) describe a polynomial time algorithm for coloring any graph with Δ + 1 colors, where Δ is the maximum degree of the graph That is, the algorithm uses the optimal number of colors for graphs of class 2, and uses at most one more color than necessary for all graphs Their algorithm follows the same strategy as Vizing's original proof of his theorem: it starts with an uncolored graph, and then repeatedly finds a way of recoloring the graph in order

to increase the number of colored edges by one More specifically, suppose that uv is an

uncolored edge in a partially colored graph The algorithm of Misra and Gries may be interpreted

as constructing a directed pseudoforest P (a graph in which each vertex has at most one outgoing

edge) on the neighbors of u: for each neighbor p of u, the algorithm finds a color c that is not used by any of the edges incident to p, finds the vertex q (if it exists) for which edge uq has color

c, and adds pq as an edge to P If the pseudoforest P constructed in this way contains a path from

v to a vertex w that has no outgoing edges in P, then there is a color c that is available both at u

and w Recoloring edge uw with color c allows the remaining edge colors to be shifted one step along this path: for each vertex p in the path, edge up takes the color that was previously used by the successor of p in the path This leads to a new coloring that includes edge uv If, on the other hand, the path starting from v in the pseudoforest P leads to a cycle, let w be the neighbor of u at which the path joins the cycle, let c be the color of edge uw, and let d be a color that is not used

by any of the edges at vertex u Then swapping colors c and d on a Kempe chain either breaks

the cycle or the edge on which the path joins the cycle, leading to the previous case With some simple data structures to keep track of the colors that are used and available at each vertex, the

construction of P and the recoloring steps of the algorithm can all be implemented in time O(n), where n is the number of vertices in the input graph Since these steps need to be repeated m

times, with each repetition increasing the number of colored edges by one, the total time is

O(mn) In an unpublished technical report, Gabow et al (1985) claimed a faster

time bound for the same problem of coloring with Δ + 1 colors

For multigraphs, Karloff & Shmoys (1987) present the following algorithm, which they attribute

to Eli Upfal Make the input multigraph G Eulerian by adding a new vertex connected by an edge

to every odd-degree vertex, find an Euler tour, and choose an orientation for the tour Form a

bipartite graph H in which there are two copies of each vertex of G, one on each side of the bipartition, with an edge from a vertex u on the left side of the bipartition to a vertex v on the

right side of the bipartition whenever the oriented tour has an edge from u to v in G Apply a bipartite graph edge coloring algorithm to H Each color class in H corresponds to a set of edges

in G that form a subgraph with maximum degree two; that is, a disjoint union of paths and cycles, so for each color class in H it is possible to form three color classes in G The time for the algorithm is bounded by the time to edge color a bipartite graph, O(m log Δ) using the algorithm

of Cole, Ost & Schirra (2001) The number of colors this algorithm uses is at most ,

close to but not quite the same as Shannon's bound of It may also be made into a parallel algorithm in a straightforward way In the same paper, Karloff and Shmoys also present a linear time algorithm for coloring multigraphs of maximum degree three with four colors (matching both Shannon's and Vizing's bounds) that operates on similar principles: their algorithm adds a new vertex to make the graph Eulerian, finds an Euler tour, and then chooses alternating sets of edges on the tour to split the graph into two subgraphs of maximum degree two The paths and even cycles of each subgraph may be colored with two colors per subgraph After this step, each remaining odd cycle contains at least one edge that may be colored with one of the two colors belonging to the opposite subgraph Removing this edge from the odd cycle leaves a path, which may be colored using the two colors for its subgraph

A greedy coloring algorithm that considers the edges of a graph or multigraph one by one, assigning each edge the first available color, may sometimes use as many as 2Δ − 1 colors, which may be nearly twice as many number of colors as is necessary However, it has the

advantage that it may be used in the online algorithm setting in which the input graph is not known in advance; in this setting, its competitive ratio is two, and this is optimal: no other online algorithm can achieve a better performance.[8] However, if edges arrive in a random order, and the input graph has a degree that is at least logarithmic, then smaller competitive ratios can be achieved.[9]

Several authors have made conjectures that imply that the fractional chromatic index of any multigraph (a number that can be computed in polynomial time using linear programming) is within one of the chromatic index.[10] If these conjectures are true, it would be possible to

compute a number that is never more than one off from the chromatic index in the multigraph case, matching what is known via Vizing's theorem for simple graphs Although unproven in general, these conjectures are known to hold when the chromatic index is at least ,

as can happen for multigraphs with sufficiently large multiplicity.[11]

It is straightforward to test whether a graph may be edge colored with one or two colors, so the first nontrivial case of edge coloring is testing whether a graph has a 3Δ/2-edge-coloring As

Kowalik (2009) showed, it is possible to test whether a graph has a 3Δ/2-edge-coloring in time O(1.3Δ/244n), while using only polynomial space Although this time bound is exponential, it is significantly faster than a brute force search over all possible assignments of colors to edges Every biconnected 3Δ/2-regular graph with n vertices has O(2n/2) 3Δ/2-edge-colorings; all of which can

be listed in time O(2n/2) (somewhat slower than the time to find a single coloring); as Greg Kuperberg observed, the graph of a prism over an n/2-sided polygon has many colorings,

showing that this bound is tight.[12]

By applying exact algorithms for vertex coloring to the line graph of the input graph, it is

possible to optimally edge-color any graph with m edges, regardless of the number of colors

needed, in time 2m mO(1) and exponential space, or in time O(2.2461m) and only polynomial space (Björklund, Husfeldt & Koivisto 2009)

Because edge coloring is NP-complete even for three colors, it is unlikely to be fixed parameter tractable when parametrized by the number of colors However, it is tractable for other

parameters In particular, Zhou, Nakano & Nishizeki (1996) showed that for graphs of treewidth

w, an optimal edge coloring can be computed in time O(nw(6w) w(w + 1)/2), a bound that depends

superexponentially on w but only linearly on the number n of vertices in the graph.

Nemhauser & Park (1991) formulate the edge coloring problem as an integer program and describe their experience using an integer programming solver to edge color graphs However, they did not perform any complexity analysis of their algorithm

[edit] Additional properties

The uniquely 3-colorable generalized Petersen graph G(9,2) One of its three color classes is shown by the light edges and the other two can be found either by rotating the light edges by 40° in each direction

or by partitioning the dark Hamiltonian cycle into alternating edges.

A graph is uniquely k-edge-colorable if there is only one way of partitioning the edges into k

color classes, ignoring the k! possible permutations of the colors For k ≠ 3Δ/2, the only uniquely

k-edge-colorable graphs are paths, cycles, and stars, but for k = 3Δ/2 other graphs may also be

uniquely k-edge-colorable Every uniquely 3Δ/2-edge-colorable graph has exactly three Hamiltonian

cycles (formed by deleting one of the three color classes) but there exist 3Δ/2-regular graphs that have three Hamiltonian cycles and are not uniquely 3Δ/2-colorable, such as the generalized Petersen