Edge-Coloring of Graphs On the left we see a 1- factorization of  5, the five-sided prism. Each factor is respresented by its own color. No edges of the same color are incident with the same vertex. In other words, the set of edges of the same color is independent. This idea can be formalized.

Edge-Coloring of Graphs A mapping c:E(G)  C from the edges set E(G) to some finite set C is called an (admissible) edge coloring, if eny two edges e and f with a common endvertex c(e)  c(f). The least number of colors neede to properly color the edges of G is called c chromatic index and is denote by  ’(G).

Vizing’s Theorem Theorem: The chromatic index of a simle graph G satisfies the following inequalities:  (G)   ’(G)   (G)+1 Proof. Lower bound immediate, the upper bound more difficult to prove. Using Vizing’s theorem we may now classify simple graphs into two types. A graph of type I has  ’(G) =  (G), while the graph of type II has  ’(G) =  (G)+1.

Consequences Corollary: Regular graph is of type I, if and only if it has a 1-factorization. Example: Petersen graph is of type II. Even if we replace a vertex by a triangle in Petersen graph, the graph remains of type II.

Triangle Removal Let G and G’ be two trivalent graphs that differ only by a triangle. This means that G’ is obtained from G by replacing any vertex by a triangle as shown on the left. Equivalently, G is obtained from G’ by a triangle removal. Proposition. If G and G’ differ by a triangle then  ’(G) =  ’(G’). a b c a b c Vertex is replaced by triangle.

Snarks Using similar argument we may prove that cycles of length 4 can be removed in the same sense. A 3-connected trivalent graph of girth > 4 of type II is called a snark.

Families of Snarks There are several infinite families of snarks known. Blanuša found the first snark after Petersen.

Blanuša snarks

König Theorem Theorem (König): For a bipartite graph G we have  ’(G) =  (G).

Exercises N1. Show that there are no bipartite snarks. N2. Repeatedly remove all quadrilaterals and triangles in K 3,3. What simple graph is obtained in the end? N3. Dot product!

Vertex Coloring of Graphs. Mapping c:V(G)  C from the vertex set to a finite set of colors is called vertex coloring, if for any pair of adjacent vertices u ~ v we have c(u)  c(v). The least number of colors of some proper vertex coloring of G is called the chromatic number and is denoted by  (G).

Brooks Theorem For any connected graph G the following holds:  (G) =  (G)+ 1, if G is isomorphic to a complete graph or an odd cycle.  (G) ·  (G), otherwise.

Four Color Theorem A theorem posed in the 19th century and proved in the 20th century. Theorem:  (G) · 4, for any planar graph G.

Graph Mappings (Homomorphisms) Let  :V(G) ! V(H) be a mapping between the vertices of two graphs.  is called a graph mapping or graph homomorphism if for any pair of vertices u,v 2 V(G) the fact u ~ v implies  (u) ~  (v).

More general maps – Weak homomorphism Sometimes we allow graph maps that do not preserve dimension.  : V(G) ! V(H) and u ~ v implies  (u) ~  (v) or  (u) =  (v). Such \phi is called a weak homomorphism.

Retracts Let  : G ! G be a graph homorphism. Then  is called a retraction if  2 = . (idempotent).  (G) = H has the property that  | H = id. H is called a retract. In a similar way one can define a weak retraction and weak retract.

Colorings revisited. A vertex coloring c with h colors can be defined as a graph homomorphism c:G ! K h

Exercises N1. Prove that retracts and weak retracts are isometric subgraphs. N2(*). A graph G is called a median graph if for any triple of vertices u,v, w we have |I(u,v) Å I(v,w) Å I(w,u)| = 1. Prove that G is a median graph if and only if it is retract of some hypercube.