1. k-Coloring k-coloring: A k-coloring of a graph G is a labeling f: V(G)  S, where |S|=k. The labels are colors; the vertices of one color form a color class. Proper k-coloring: A k-coloring is proper if adjacent vertices have different labels. k-colorable graph: A graph is k-colorable if it has a proper k-coloring. Chromatic number  (G): The least k such that G is k-colorable.  (G)=3 3-colorable graph. 2-colorable graph?

2. Example 5.1.3 Petersen Graph: The Petersen graph is the simple graph whose vertices are the 2-element subsets of a 5- element set and whose edges are the pairs of disjoint 2-element subsets. A graph is 2-colorable if and only if it is bipartite.  C5 and Petersen graph have chromatic number at least 3.

3. k-chromatic graph k-chromatic graph: A graph G is k-chromatic if  (G)=k. A proper k-coloring of a k-chromatic graph is an optimal coloring. k-critical graph : If  (H)<  (G)=k for every proper subgraph H of G, then G is k-critical. Clique Number: The clique number of a graph G, written  (G), is the maximum size of clique in G. cd ab ef  (G)=4 3-critical graph

4. Proposition 5.1.7 For any graph G,  (G)>=  (G) and  (G)>=n(G)/  (G). Proof.  (G)>=  (G).  2.  (G)>=n(G)/  (G).  Vertices of a clique requires distinct colors. At most  (G) vertices can have the same color.

5. Example 5.1.8 of  (G)>  (G) 1. For r>=2, let G=C 2r+1  K s. 2. C 2r+1 has no triangle   (G)=s+2. 3. C 2r+1 needs at least three colors, say a, b, and c. 4. K s needs s colors which must differ from colors a, b, and c.   (G)>=s+3. 5.  (G)>  (G).

6. Greedy Coloring The greedy algorithm relative to a vertex ordering v 1, v 2, …, v n of V(G) is obtained by coloring vertices in the order v 1, v 2, …, v n, assigning to v i the smallest- indexed color not already used on its lower-indexed neighbors. 1 2 3 4 5 6 1 2 3 4 5 6 1 2 4 3

7. Proposition 5.1.13  (G)<=  (G)+1. Proof. 1. In a vertex ordering, each vertex has at most  (G) earlier neighbors.  Greedy coloring cannot be forced to use more than  (G)+1 colors.

8. Block Block: A maximal connected subgraph of G that has no cut-vertex. If G itself is connected and has no cut- vertex, then G is a block. (Definition 4.1.16)

9. Block-cutpoint graph Block-cutpoint graph: The block-cutpoint graph of a graph G is a bipartite graph H in which one partite set consists of the cut-vertices of G, and the other has a vertex b i for each block B i of G. vb i as an edge of H if and only if v  B i. b1b1 b5b5 b3b3 b2b2 b4b4

10. Leaf Block Leaf Block: A block that contains exactly one cut- vertex of G. When G is connected, its block-cutpoint graph is a tree (Exercise 34 of Sec. 4.1) whose leaves are blocks of G.  A graph that is not a single block has at least two leaf blocks. b1b1 b5b5 b3b3 b2b2 b4b4

11. Brook’s Theorem If G is a connected graph other than a complete graph or an odd cycle, then  (G)<=  (G). Proof. 1. Let k=  (G). 2. Since G is a complete graph when k =3. 3. The theorem holds if we can order the vertices such that each has at most k-1 lower-indexed neighbors.

12. Brook’s Theorem 4. Case 1: G is not k-regular. Let v n be the vertex of degree less than k. 5. Grow a spanning tree of G from v n, assigning indices in decreasing order as we reach vertices. 1 2 3 4 5 6 1 2 4 3 12 3 4 5 6 6. Each vertex other than v n in the resulting ordering has v 1, v 2, …, v n has a higher- indexed neighbor along the path to v n in the tree.  Each vertex has at most k-1 lower-indexed neighbors.

13. Brook’s Theorem 4. Case 2: G is k-regular. 1 2 3 4 5 6 1 2 3 4 5 x G’ 5. Case 2-1: G has a cut-vertex x. 6. Let G’ be a subgraph consisting of a component of G- x together with its edges to x. 7. The degree of x in G’ is less than k.  The method in case 1 provides a proper k-coloring of G’. G’ 8. By permuting the names of colors in the subgraphs resulting in this way from components of G-x, we can make the colorings agree on x to complete a proper k-coloring of G.

14. Brook’s Theorem 9. Case 2-2: G is 2-connected. 10. Suppose that some vertex v n has neighbors v 1, v 2 such that (v 1,v 2 )  E(G) and G-{v 1,v 2 } is connected. 11. Index the vertices of a spanning tree of G-{v 1, v 2 } using 3, 4, …, n such that labels increase along paths to the root v n. 12. Each of v 1, v 2, …, v n has at most k-1 lower indexed neighbors. 13. v 1 and v 2 receives the same color.  At most k-1 colors are used on neighbors of v n.

15. Brook’s Theorem 14. It suffices to show that every 2-connected k-regular graph with k>=3 has such a triple v 1,v 2,v n in 10. 15. Choose a vertex x. 16. Case 2-2-1:  (G-x)>=2. 17. Let v 1 be x. 18. There exists a vertex v 2 with distance 2 from x because G is not a complete graph and G is regular. 19. Let v n be a common neighbor of v 1 and v 2. 20. v 1, v 2, v n be the desired triple. 21. Case 2-2-2:  (G-x)=1.

16. Brook’s Theorem 22. Let v n =x. Then, x has a neighbor in every leaf block of G-x. Otherwise, G is not 2-connected. 23. G-x is not a single block  At least two leaf blocks in G-x 24. Clearly, neighbors v 1 and v 2 of x are not adjacent. 25. G-{v 1,v 2,x} is connected since blocks have no cut- vertices. 26. k>=3.  vertex x has a neighbor other than v 1 and v 2  G-{v 1,v 2 } is connected. because  (G-x)=1. because G-x is connected.