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

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 10, Monday, September 22. 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 c ’(G). Theorem 3 (Vizing, 1964).

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

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  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 byc’(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 ·c’(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).

  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 3 2 4 8 1 6 5 7 3 2 4 8 1 6 5 7 3 2 4 8 1 6 5

  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.

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