# Pearls in Graph Theory: A Comprehensive Introduction (Dover Books on Mathematics)

## Nora Hartsfield, Gerhard Ringel

Language: English

Pages: 201

ISBN: B01JXOXFEG

Format: PDF / Kindle (mobi) / ePub

Based on 20 years of teaching by the leading researcher in graph theory, this text offers a solid foundation on the subject. Topics include basic graph theory, colorings of graphs, circuits and cycles, labeling graphs, drawings of graphs, measurements of closeness to planarity, graphs on surfaces, and applications and algorithms. 1994 edition.

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For every vertex z in V – W we have degG(z) ≤ degG(x), since x was the vertex of maximal degree in G, and degG(x) = degH(x) = degH(z), because we connected every vertex in V – W to every vertex in W to construct H. Thus for every vertex z in V – W degG(z) ≤ degH(z). Now let z be a vertex in W. Let w be the number of vertices in W. Then degG(z) ≤ degGo(z) + n − w, since z can be adjacent to at most n – w of the vertices in V – W in G. Also, degGo(z) + n – w ≤ degHo(z) + n – w by the

1-factor shown in Figure 5.1.1, the resulting graph will have as 1-factors only those 1-factors of K3,3 that contain no vertical edge. Thus D3 = 2. The dotted lines indicate that the edge is not supposed to be used. We can derive an expression for Dn in terms of Dn−1 and Dn−2. First, we count the number of 1-factors that contain the edges 1 k′ and k1′. There are Dn−2 such 1-factors for each value of k, and k can assume n−1 values. Thus this gives us (n−1)Dn−2 possibilities. Now we count

c by the following recipe. If an edge lies between countries colored color the edge l and 2 a 3 and 4 a 1 and 3 b 2 and 4 b 1 and 4 c 2 and 3 c. Figure 8.2.3 shows the recipe. The edges are now colored by three colors. We must show that this is a proper coloring. Figure 8.2.3 In Figure 8.2.4 we have all the ways, up to turnings and reflections, that three countries in a normal map colored with four colors can meet at a vertex. In each case, the recipe gives

Theorem 8.2.5. If in a normal map M, the vertices can be labeled by black and white so that around each country in M the number of black vertices minus the number of white vertices is a multiple of three, then the edges of M can be properly colored by three colors. Proof. Given a normal map M and a labeling of the vertices by black and white with the above property, we construct the normal M ’ by blowing up every white vertex into a triangle with three black vertices. So we have a map M ‘ such

p such that there are four induced circuits and their lengths are 3, 4, 5, and 6. (Hint: There might be vertices of degree 2.) 10.1.6.Prove that it is not possible to find a graph G with a rotation p such that the lengths of the five induced circuits are 3, 4, 5, 5, and 6. 10.1.7.How many different rotations does a cubic graph with 2? vertices have? 10.1.8.Suppose that a graph G has p vertices with degrees d1,d2,d3, ... dp. How many different rotations does G have? 10.1.9.List the circuits