By John M. Harris, Jeffry L. Hirst, Michael J. Mossinghoff (auth.)

ISBN-10: 1475748035

ISBN-13: 9781475748031

ISBN-10: 1475748051

ISBN-13: 9781475748055

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**Extra info for Combinatorics and Graph Theory**

**Example text**

22. 23. For any graph G. (G) is the maximum degree of G. (G) + 1 colors. (G) + 1. (G) + 1. 0 PROOF. Notice that we obtain equality in this bound for complete graphs and for cycles with an odd number of vertices. 23 holds. This is stated in Brooks's Theorem [ 17]. The proof that we give is a modification of the one given by Lovasz [ 102}. 24 (Brooks's Theorem). ( G). Let G be a connected graph of order n that is neither a complete graph nor an odd cycle. (G). We know that k # 0 and k # 1, since otherwise G is complete.

C. d. e. 2 Find all 1-critical and 2-critical graphs. Give an example of a 3-critical graph. If G is k-critical, then show that G is connected. If G is k-critical, then show that 8(G) 2: k- 1. Find all of the 3-critical graphs. Hint: Use part (d). Bounds on Chromatic Number The point is, ladies and gentlemen, that greed, for lack of a better word, is good. Greed is right. Greed works. - Gordon Gekko, in Wall Street In general, determining the chromatic number of a graph is hard. While small or well-known graphs (like the ones in the previous exercises) may be fairly easy, the number of possibilities in large graphs makes computing chromatic numbers difficult.

The removal of v from G will form at least two connected components. Say the components are G1> Gz, ... , Gr. 61). H 1 is a connected graph, and the degree of v in H is less thank. 61. The graph H1. 50 l. Graph Theory with at most k colors. Similarly, we can properly color each Hi = G; -- {v} with at most k colors. Without loss of generality, we can assume that v gets the same color in all of these colorings (if not, just permute the colors to make it so). These colorings together create a proper coloring of G that uses at most k colors.

### Combinatorics and Graph Theory by John M. Harris, Jeffry L. Hirst, Michael J. Mossinghoff (auth.)

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