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.

Example 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.

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.

Proposition 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).

Example of  (G)>  (G) 1. For r>=2, let G=C 2r+1  K s. 2. C 2r+1 has no triangle   (G)=s 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.

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.

Proposition  (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.

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.

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.

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.

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.

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 Choose a vertex x. 16. Case 2-2-1:  (G-x)>= 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 v 1, v 2, v n be the desired triple. 20. Case 2-2-2:  (G-x)=1.

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.

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.

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.