3. [10 marks] Let Go = (V,E) and G₁ = (V,E₁) be two graphs on the same set of vertices. Let (V, EU E1), so that (u, v) is an edge of H if and only if (u, v) is an edge of Go or of G1 (or of both). H = (a) Show that if Go and G₁ are both Eulerian and En E₁ = Ø (i.e., Go and G₁ have no edges in common), then H is also Eulerian. (b) Give an example where Go and G₁ are both Eulerian, but H is not Eulerian.

3 and 4

△ABC has vertices A(0, 0), B(2, 4), and MC(4, 2). △RST has vertices R(0, 3), S(-1, 5), and T(-2, 4). Prove that △ABC is similar to △RST.

Show that For n > 1 let Gn be the simple graph with vertex set V(Gn) = {1,2, ., n} in which two different vertices i and j are adjacent whenever j is a multiple of i or i is a multiple of j. For what n is Gn planar? ...1

8. Graph G has vertex set V (G) = {u1, U2, U3, U4, U5, U6} and edge set E(G) = {e1, €2, e3, e4, e5, e6} with edge-endpoint function Edge Endpoints {u1, U2} {u3, U4} {u3, U5} {u1, U3} {u1, U4} {u4, U5} e1 e2 e3 e4 e5 e6 (a) Sketch G. Be sure to label the vertices and edges: (b) Find the degrees of each of the vertices of G.

i) Consider the set S = {−6, −3, 0, 3, 6}. Draw a graph G whose set of verti- ces be S and such that for i, j ∈ S, ij ∈ E(G) if ij are related to a rule that t'u you choose to apply to i and j. (ii) A graph G of order 12 has as a set of vertices c1, c2, . . . , c12 for the do- ce configurations of figure 1. A movement on said board corresponds to moving a coin to an unoccupied square using the following two rules: 1. the gold coin can move only horizontally or diagonally, 2. the silver coin can move only vertically or diagonally. Two vertices ci, cj, i̸ = j are adjacent if it is possible to move ci to cj in a single movement. a) What vertices are adjacent to c1 in G? b) Draw the subgraph induced by {c2, c6, c9, c11}

7. Triangle JKL with vertices J(-3, 1), K(–1, 4), and L(6, –5) | -5 5. -5

Find k so that the points P(0, 1, 1), Q(-1, 1, 3), and R(-3, 1, k) are on the same line (collinear).

3. Let G be a siugle graph with vertices M,V2, Vs, V4 aud Vg. Let 0101 10I0I 01011 A - be ts adjacaney matrix. ( Find all paths of leugth 3 be tween Vq and V (b) V and Vu It B. amazon pe here to search

17.) What is the vertex of y = - | x | - 2? a. (0, -2) b. (0,2) c. (2, 0) d. (2, -2)