Coloring Basics A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]). For an i S, the set f 1 (i) is called a color class. A k-coloring is called proper if adjacent vertices receive different colors.
A graph is called k-colorable if it has a proper k-coloring. The chromatic number of G, χ(g), is the least k such that G admits a proper k-coloring. Fact 1 For every graph G on n vertices, χ(g) ω(g), n α(g) χ(g) (G) + 1.
Mycielski s construction Theorem 2 For every positive integer k there exists a simple graph G k with ω(g k ) = 2 and χ(g k ) = k + 1. Mycielski s construction: Given a simple graph G k with vertex set V = {v 1,..., v n } consider G k+1 with vertex set V U {w} where U = {u 1,..., u n } and edges defined as follows. Edges between V are as in G, for every u i, add edge u i v j whenever v j N(v i ), and add wu i for every i.
Proof. Induction on k. =
Brooks theorem Lemma 3 Let G be a simple graph on n vertices and let v V (G). There exists an ordering v 1,..., v n of V (G) such that v = v n and for every i < n there is j > i such that v i v j E(G). A b-obstruction: K b, or, if b = 3 and odd cycle. Let ω (G) be the largest integer b such that G has a b-obstruction. We have χ(g) ω (G).
Theorem 4 (Brooks) For every simple graph G, χ(g) max{ω (G), (G)}. Corollary 5 If G is a connected graph with maximum degree which is neither K +1 nor an odd cycle, then χ(g).
Proof. y P w w' Q z=w' x y w w' Q Interchange w with w' x P P is a y-q path in G-x, y,,z and z is not w w y z x P- y Q- w
Turáns theorem A graph is called r-partite if it is r-colorable. Complete r-partite graph K n1,n 2,...,n r. Turán graph T n,r - complete r-partite graph on n vertices in which sizes of any two parts differ by at most one.
Lemma 6 Among all r-partite graphs on n vertices T n,r is the unique graph with a maximum number of edges. Theorem 7 (Turán) Among all graphs on n vertices with no r+1-clique, T n,r is the unique graph with the maximum number of edges. Proof. N(v)
Edge Coloring A proper k-edge-coloring of a (multi)graph G is a function f : E(G) {1,..., k} such that f(e) = f(e ) implies e e =. The chromatic index of G, χ (G), is the least k such that G admits a proper k-edge-coloring. Fact 8 (König) If G is a bipartite multigraph then χ (G) = (G).
Proof. May assume the partite sets have the same size. May assume G is (G)-regular.
Lemma 9 Let G be a simple graph with (G) k and let v V. If χ (G v) k and d(x) = k for at most one x N(v), then χ (G) k. Proof. f(x) - the set of colors not present in edges incident to x f α - the set of vertices x N(v) that contain α in f(x). Let y be the unique vertex in N(v), d(y) = k. Assume that the remaining have degree k 1.
Choose a k-edge-coloring f of G v maximizing T(f) := {β 1 f β 2}. Case 1: For every color α, f α = 1 There exist two colors β, γ such that f γ 3, f β = 0. Consider G β,γ and exchange colors on a path starting at w f γ to contradict the maximality of T(f). Case 2: f α = {z}, say α = k. Let M := f 1 (k)+vz and note that M is a perfect matching in N(v) {v}.
Let H = G M. Then d H (x) k 1 for every x N H (v) with equality holding at most once. In addition, f is a (k 1)-edge-coloring of H v and so (H v) k 1. Thus (H) k 1 and we are done by induction. Theorem 10 (Vizing) For every simple graph G, χ (G) (G)+ 1.
List Coloring Definition 1 For each vertex v let L(v) be the list of colors available at v. A list coloring is a proper vertex coloring f such that for every v, f(v) L(v). A graph is k-choosable if every assignment of list of size k admits a proper coloring. The choice number, χ l (G), is the least k such that G is k-choosable. Fact 11 If m = ( ) 2k 1 k then χl (K m,m ) > k. Proof. Assign distinct k-element subsets of [2k 1] as lists to vertices from X and the same to Y.
Case 1: Coloring uses less than k colors on X. Case 2: Coloring uses at least k colors on X. Definition 2 For each edge e let L(e) be the list of colors available at a. A list edge-coloring is a proper edge-coloring f such that for every e, f(e) L(e). A graph is k-edge-choosable if every assignment of list of size k admits a proper edge-coloring. The list chromatic index, χ l (G) is the least k such that G is k-edge-choosable. Conjecture 1 (List Coloring Conjecture) For a graph G, χ l (G) = χ (G).
Definition 3 A kernel of a digraph is an independent set S having a successor of every vertex of V S. A digraph is called kernel-perfect if every induced sub-digraph has a kernel. Given a function f : V (G) N, the graph is f-choosable if for any list assignment the with L(v) = f(v) there is a proper vertex coloring c, such that for every v, c(v) L(v).
Lemma 12 (Kernel Lemma) Let f : V (G) Z +. If a graph G has an orientation D such that D is kernel-perfect and d + (x) < f(x) for all x V (G) then G is f-choosable. Theorem 13 (Galvin) For every bipartite graph G, χ l (G) = (G). Proof. Consider the line graph H := L(G). There is a vertex coloring c : V (H) {1,..., (G)}. Consider the following orientation of the edges of H. If ee E(H) and e e X then use the orientation e e if c(e) > c(e ) and e e otherwise.
If ee E(H) and e e Y then use the orientation e e if c(e) < c(e ) and e e otherwise. Claim 14 d + (e) < (G). Claim 15 Every induced sub-digraph D of the above orientation D of H has a kernel. Induction on V (D ). Let x X, e x V (D ) contains x and has the smallest color. Let U := {e x x X}. Case 1: U is independent.
Case 2: e x, e x U, e x e x = {y}. Say e x e x. Consider D := D e x. x e* e(x ) Induction to D e(x) x e(x) By induction D has a kernel U. If e x / U, then e U such that e x e. Then e x e.