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. • 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.

2. 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

3. 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.

4. 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)

5. 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

6. 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

7. 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.

8. 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

9. 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

10. 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.

11. 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.

12. 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.

13. Definition 7.1.3 • k-edge-coloring of G: A labeling f: E(G) -> S, where |S| =k. The labels are colors; the edges of one color form a color class. • proper k-edge-coloring: A k-edge-coloring such that incident edges have different labels. • k-edge-colorable graph: A graph having a proper k-edge-coloring. • edge-chromatic number, x’(G), of G: The least k such that G is k-edge-colorable.

14. Theorem 7.1.10 • If G is a simple graph, then x’(G) ≤Δ(G)+1. • Proof: 1.Let f be a proper Δ(G)+1-edge-coloring of a subgraph G’ of G. • 2. If G’≠ G, then some edge uv is uncolored by f. • 3. It suffices to show we can extend the coloring to include uv after possibly recoloring some edges (an augmentation).

15. a0 u a1 v1 v0= v a1 Theorem 7.1.10 4.Every vertex has some color not appearing on its incident edges because the number of colors exceeds Δ(G). 5. Let a0 be a color missing at u. 6. Let a1 be a color missing at v0(=v). We may assume that a1 appears at u on some edge uv1; otherwise, we would use a1 on uv0.

16. a0 a3 u v2 a2 a1 a2 a1 v1 v0= v Theorem 7.1.10 7. Let a2 be a color missing at v1. We may assume that a2 appears at u on some edge uv2; otherwise, we would replace color a1 with a2 on uv1 and then use a1 on uv0 to augment the coloring.

17. ak+1 a1 Theorem 7.1.10 8. Having selected uvi-1 with color ai-1, let ai be a color missing at vi-1. If ai is missing at u, then we use ai on uvi-1 and shift color aj from uvj to uvj-1 for 1≤ j ≤i-1 to complete the augmentation. We call this downshifting from vi-1. If ai appears at u (on some edges uvi), then the process continues. ai ai-1 u vi-1 ak vk v0= v

18. ak al u a0 vl ak ak+1 ak ak-1 a1 vk vk-1 v0= v Theorem 7.1.10 9. Since we have only Δ(G)+1 colors to choose from, the list of selected colors eventually repeats (or we complete the augmentation by downshifting). 10. Let l be the smallest index such that a color missing at vl is in the list a1,…,al; let this color be ak. 11. The color ak missing at vl is also missing at vk-1 and appears on uvk. If a0 does not appear at vl, then we downshift from vl and use color a0 on uvl to complete the augmentation. Hence we may assume that a0 appears at vl.

19. Theorem 7.1.10 12. Let P be the maximal alternating path of edges colored a0 and ak that begins at vl along color a0. There is only one such path because each vertex has at most one incident edge in each color (we ignore edges not yet colored).

20. Theorem 7.1.10 13. If P reaches vk, then it arrives at vk along an edge with color a0, follows vku in color ak, and stops at u, which lacks color a0. In this case, we downshift from vk and switch colors on P.

21. Theorem 7.1.10 14. If P reaches vk-1, then it reaches at vk-1 on color a0, and stops there, because ak does not appear at vk-1. In this case, we downshift from vk-1, give color a0 to uvk-1, and switch colors on P.

22. Theorem 7.1.10 15. If P does not reach vk or vk-1, then it ends at some vertex outside {u, vl, vk, vk-1}. In this case, we downshift from vl, give color a0 to uvl, and switch colors on P.