1.2.9. (-) What is the minimum number of trails needed to decompose the Petersen graph? Is there a decomposition into this many trails using only paths?

The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n > 3, and connect this new vertex to each of n vertices by new edges. Match between each statement (a)- (d)) and a graph ((1)-(5)) such that the chosen graph satisfies the statement. (1) C, (2) C10 (3) W, (4) W, (5) W 10 A graph with the sum of degrees is 28. Choose... A simple and bipartite graph Choose... A simple graph with Hamiltonian circuit and vertex of degree 10. Choose... : A regular graph and not bipartite graph Choose...

Kn,n,n be a complete 3-bipartite graph whose maximal independent sets are C1 = [n],C2 = [2n] – [n], and C3 = [3n] – [2n]. - (a) How many edges does Kn,n,n have? (b) Does Kn,n,n have an Eulerian cycle? (c) Show by any means that Kn,n,n has a Hamilton cycle. (d) Let G be the graph that is obtained from Kn,n,n by removing the edges of your Hamilton cycle. Does G have an Eulerian cycle?

16 Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <4 (Exercise 19 in Section 1.4) to prove that every planar graph with less than 12 vertices can be 4-colored.

(a) (5 points) Show that there is a a bijection between the set of noni- somorphic (as 3-ary trees) 3-ary trees with n vertices and the set of nonisomorphic (as 3-ary trees) full 3-ary trees with 2n + 1 terminal vertices.

Show that Z3 X Z5 is cyclic.

1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove: if all vertices have degree either 1 or at least 4, then T has at least 2(n + 1)/3 leaves. (4 pt)

1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231

. You are traveling on a grid from (2, 2) to (18, 18). The only steps you can make are either to increase the first number by 1, or to increase the second number by 1, which you can think of as unit steps that either go up or to the right. (a) Identify a bijection between the paths from (2, 2) to (18, 18) and bit sequences with a certain number of ones. Hint: there are two types of steps, so let one of these be a zero, and the other a one. (b) How many paths are there from (2, 2) to (18, 18)? (c) How many of these paths do not pass through (10, 10)? (d) How many of these paths do not pass through (7,7) or (10, 10)? (Note: this can also be phrased as paths that avoid both (7,7) and (10, 10), using De Morgan's Law to negate the or notice this is a time where "and" does not mean multiply in counting).

Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced subgraph isomorphic to k1,r+1 has a 1-factor. Show that this is not true if you replace r connected by r edge connected