PS04-100323 - which the greedy coloring uses only χ ( G )...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 682 Problem Set #4 This problem set is due at the beginning of class on March 23 . Below, “graph” means “simple finite graph” except where otherwise noted. 1. (10 points) The Petersen graph is shown below. (a) (5 points) Demonstrate that the Petersen graph is nonplanar by invoking Kura- towski’s Theorem. (b) (5 points) Demonstrate that the Petersen graph is nonplanar by invoking Euler’s theorem. 2. (10 points) Prove the following results about chromatic number: (a) (5 points) Show that on any graph G , χ ( G ) n α ( G ) . (b) (5 points) Show that if G has decomposition into blocks B 1 ,B 2 ,...,B n , then χ ( G ) = max i ( χ ( B i )). 3. (15 points) A greedy coloring of a graph with an ordered set of vertices v 1 ,v 2 ,...,v n is produced by labeling each vertex in order with the lowest number not already used at an adjacent labeled vertex. (a) (5 points) Show that, for any graph G , there is some ordering of the vertices on
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: which the greedy coloring uses only χ ( G ) colors. (b) (5 points) Demonstrate that for any k > 2 there is a bipartite graph G on 2 k vertices and vertex ordering thereon in which the greedy coloring uses k colors, even though χ ( G ) = 2. (c) (5 points) Show that a greedy coloring of the cycle C n , regardless of the value of n , does not use more than 3 colors. 4. (5 points) Prove that a bipartite graph on n > 2 vertices is planar only if it has 2 n-4 or fewer edges. 5. (5 point bonus) Prove that for every value of k > 2, there is a tree and ordering of the vertices thereon such that a greedy coloring (see above) of the tree requires k colors. What is the smallest such tree you can find? Page 1 of 1 Due March 23, 2010...
View Full Document

This note was uploaded on 01/12/2012 for the course MATH 682 taught by Professor Wildstrom during the Spring '09 term at University of Louisville.

Ask a homework question - tutors are online