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### hw5

Course: MATH 544, Fall 2009
School: CSU San Marcos
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Word Count: 369

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544, Math Dr. Kndgen u CSUSM, Spring 2006 Assignment 5 due: Thursday, April 13, 2006 1. (8.2.15/3.4.25) Mirzakhanis non-4-choosable planar graph on 63 vertices. (a) In the list assignment of the graph shown in (8.2.15a), S = [4] = {1, 2, 3, 4} and i = [4]{i}. Prove that this graph has no proper coloring from the lists. (b) In the list assignment of the graph G shown in (8.2.15b), i denotes [5] {i}. Let G be the...

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544, Math Dr. Kndgen u CSUSM, Spring 2006 Assignment 5 due: Thursday, April 13, 2006 1. (8.2.15/3.4.25) Mirzakhanis non-4-choosable planar graph on 63 vertices. (a) In the list assignment of the graph shown in (8.2.15a), S = [4] = {1, 2, 3, 4} and i = [4]{i}. Prove that this graph has no proper coloring from the lists. (b) In the list assignment of the graph G shown in (8.2.15b), i denotes [5] {i}. Let G be the graph obtained from G by adding a vertex v with list 1 adjacent to all vertices on the outside face in this drawing of G. Prove that G has no proper coloring from these lists. 2. Let G be the graph of the icosahedron, i.e. the vertices and edges are those of the platonic solid with 20 faces exercise (8.2.37) shows a nice drawing. (a) Prove that up to the names of the vertices G is the only 5-regular simple planar graph on 12 vertices. (b) (8.2.37) Prove that G has no (0,2)-partition. (c) Find all sequences of integers 0 k1 k2 . . . kr such that G has a (k1 , k2 , . . . , kr )partition. 3. Assorted examples (a) Find the minimum k such that the Petersen graph acyclically is k-colorable. (b) Find the minimum k such that the Petersen graph is k star-colorable. (c) Find the minimum k such that the 3-cube Q3 is acyclically k-colorable. (d) Find the minimum k such that the 3-cube Q3 is k star-colorable. (e) For each a 3 give an example of a graph Ga with acyclic coloring number a, so that Ga has no star-coloring with a colors, and (Ga ) < a. 4. (a) Prove that every graph that has an acyclic a-coloring has a star coloring with at most a2a1 colors without appealing to results from lecture 19. (b) Show that trees are 3-star-choosable 5. Recall that the grid graph Pm Pn is the cartesian product of two p...

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