k-Coloring

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. • C5 and Petersen graph have chromatic number at least 3.

3. a b e f 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. (G)=4 3-critical graph c d

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=C2r+1Ks. 2. C2r+1 has no triangle  (G)=s+2. 3. C2r+1 needs at least three colors, say a, b, and c. 4. Ks 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 v1, v2, …, vn of V(G) is obtained by coloring vertices in the order v1, v2, …, vn, assigning to vi the smallest-indexed color not already used on its lower-indexed neighbors. 3 2 1 4 2 5 3 2 3 5 1 4 6 4 6 1

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 bi for each block Bi of G. vbi is an edge of H if and only if v Bi. b5 b3 b1 b2 b4

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. b5 b3 b1 b2 b4

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. When k<=1, G is a complete graph. 3. When k=2, G is an odd cycle or is bipartite, in which case the bound holds. 4. We assume that k>=3. 5. The theorem holds if we can order the vertices such that each has at most k-1 lower-indexed neighbors.

12. Brook’s Theorem 6. Case 1: G is not k-regular. Let vn be the vertex of degree less than k. 5. Grow a spanning tree of G from vn, assigning indices in decreasing order as we reach vertices. 6. Each vertex other than vn in the resulting ordering has v1, v2, …, vn has a higher-indexed neighbor along the path to vn in the tree.  Each vertex has at most k-1 lower-indexed neighbors. 5 2 1 3 5 4 3 2 3 1 2 4 6 4 6 1

13. Brook’s Theorem G’ 5 4. Case 2: G is k-regular. 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. 8. The method in case 1 provides a proper k-coloring of G’. 9. 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. 3 4 1 2 x 6 5 G’ 3 4 1 2

14. Brook’s Theorem 10. Case 2-2: G is 2-connected. 11. Suppose that some vertex vn has neighbors v1, v2 such that (v1,v2)E(G) and G-{v1,v2} is connected. 12. Index the vertices of a spanning tree of G-{v1, v2} using 3, 4, …, n such that labels increase along paths to the root vn. 13. Each of v1, v2, …, vn-1 has at most k-1 lower indexed neighbors. 14. v1 and v2 receives the same color.  At most k-1 colors are used on neighbors of vn.

15. Brook’s Theorem 15. It suffices to show that every 2-connected k-regular graph with k>=3 has such a triple v1,v2,vn in 10. 16. Choose a vertex x. 17. Case 2-2-1: (G-x)>=2. 18. Let v1 be x. 19. There exists a vertex v2 with distance 2 from x because G is not a complete graph and G is regular. 20. Let vn be a common neighbor of v1 and v2. 21. v1, v2, vn is the desired triple. 22. Case 2-2-2: (G-x)=1.

16. Brook’s Theorem Otherwise, G is not 2-connected. because (G-x)=1. 23. Let vn=x. Then, x has a neighbor in every leaf block of G-x. 24. G-x is not a single block 25. At least two leaf blocks in G-x 26. Clearly, neighbors v1 and v2 of x are not adjacent. 27. G-{v1,v2,x} is connected since blocks have no cut-vertices. 27. Vertex x has a neighbor other than v1 and v2 because k>=3. 28. G-{v1,v2} is connected. because G-x is connected.