Graph Theory

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

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

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Extra resources for Combinatorics and Graph Theory

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20. A graph G is planar if and only if every subdivision ofG is planar. 53. 54. A graph and a subdivision. 3 or Ks as subgraphs, and all graphs containing a subdivision of K 3 . 3 or K 5 . The list so far stems from only two specific graphs: K 3 _3 and K 5 . A well-known theorem by Kuratowski [99] tells us that there are no other graphs on the list! The bottom line is that K 3 . 3 and K 5 are the only two real enemies of planarity. Kuratowski proved this beautiful theorem in 1930, closing a long-open problem.

Let T be a labeled tree. Prove that the Priifer sequence ofT will not contain any of the leaves' labels. Also prove that each vertex v will appear in the sequence exactly deg( v) - 1 times. 2. 42. 3. Draw and label a tree whose Priifer sequence is 5,4,3,5,4,3,5,4,3. 4. Which trees have constant Priifer sequences? 5. Which trees have Priifer sequences with distinct terms? 6. Let e be an edge of Kn. Use Cayley's Theorem to prove that Kn - e has (n - 2)nn- 3 spanning trees. 7. Use the Matrix Tree Theorem to prove Cayley's Theorem.

11 (Matrix Tree Theorem). If G is a connected labeled graph with adjacency matrix A and degree matrix D, then the number of unique spanning trees of G is equal to the value of any cofactor of the matrix D - A. Suppose G has n vertices (v 1 , •.. , vn) and kedges (fl, ... , fk). Since G is connected, we know that k is at least n - 1. Let N be the n x k matrix whose i, j entry is defined by PROOF. [N]·. = '·l 1 1 if v; and h are incident, 0 otherwise. N is called the incidence matrix of G. Since every edge of G is incident with exactly two vertices of G, each column of N contains two 1's and n - 2 zeros.

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