Download presentation

Presentation is loading. Please wait.

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.

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.

4
Proposition 5.1.7 For any graph G, (G)>= (G) and (G)<=n(G)/ (G). Proof. 1. Vertices of a clique requires distinct colors. (G)>= (G). 2. Each color class is an independent set. (G)<=n(G)/ (G).

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.

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.

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
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.

9
Brook’s Theorem (2/6) 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. 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.

10
Brook’s Theorem (3/6) 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. The method in case 1 provides a proper k-coloring of 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.

11
Brook’s Theorem (4/6) 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.

12
Brook’s Theorem (5/6) 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 and let v 2 be a vertex with distance 2 from x. 18. Let v n be a common neighbor of v 1 and v 2. 19 v 1, v 2, v n be the desired triple. 20. Case 2-2-2: (G-x)=1.

13
Brook’s Theorem (6/6) 21. Let v n =x. Then, x has a neighbor in every leaf block of G-x. Otherwise, G is not 2-connected. 22. G-x is not a single block. At least two leaf blocks in G-x. 23. Clearly, neighbors v 1 and v 2 of x are not adjacent. 24. G-{v 1, v 2, x} is connected since blocks have no cut- vertices. 25. k>=3. vertex x has a neighbor other than v 1 and v 2 G-{v 1, v 2 } is connected.

14
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.

15
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 single block has at least two leaf blocks.

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google