1. MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 10, Monday, September 22

2. Coloring Edges On the left we see an edge coloring of a graph. The minimum number of colors needed in such a coloring is called the edge chromatic number and is denoted by ’(G).

3. Theorem 3 (Vizing, 1964) If the maximum degree of a vertex in a graph G is d, then the edge chromatic number of G is either d or d+1. In other words: d · ’(G) · d+1.

4. Theorem 4 Every planar graph can be 5- colored.

5. 3.1 Properties of Trees Homework (MATH 310#4M): Read 3.2. Do Exercises 3.1: 2,4,6,10,12,14,16,18,24,30 Volunteers: ____________ Problem: 30. On Monday you will also turn in the list of all new terms (marked). On Monday you will also turn in the list of all new terms (marked).

6. What is a Tree? There are at least three ways to define a tree. We will distinguish the following: tree rooted tree ordered (rooted) tree [will not be used] 7 5 48 1 6 23 7 5 48 1 6 23 7 5 48 1 6 23

7. A Tree A tree is a connected graph with no circuits. There are several characterizations of trees; compare Theorem 1, p.96 and Exercise 5, p.102. For example: A tree is a connected graph with n vertices and n-1 edges. A tree is a graph with n vertices, n-1 edges and no circuits. A tree is a connected graph in which removal of any edge disconnects the graph. A tree is a graph in which for each pair of vertices u and v there exists an unique path from u to v.

8. A Spanning Tree Each connected graph has a spanning tree. For finite graphs the proof is easy. [Keep removing edges that belong to some circuit]. For infinite graphs this is not a theorem but an axiom that is equivalent to the renowned axiom of choice from set theory. Note: A spanning subgraph H of G contains all vertices of G.