7. Let G be a loop-free undirected graph, where ∆ = maxv∈V {deg(v)}.a) Prove that χ(G) ≤ ∆ + 1.b) Find two types of graphs G, where χ(G) = ∆ + 1.

Call a graph H regular if there exists k ≥ 1 such that dµ(v) = k for all v € V(H). (e) Does there exist a regular graph H with V(H) = V(G) and E(H) ≤ E(G)? Justify your answer.

Let F, F' be forests on the same set of vertices, with ||F|| < ||F'||. Show that F' has an edge e such that F + e is again a forest.

3. Let A = {1, 2, 3}, let S = P(A) and let RC S x S be a relation defined by the rule (X,Y) ER⇒ (XnY=0)A (XUY = A). (a) Prove or find a counterexample for the following properties: · antisymmetry. asymmetry. • symmetry. ● irreflexivity/ • reflexivity. • transitivity • uniqueness (b) Based on the findings from the previous point, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.

Q4. Prove that a complete bipartite graph Km,n is Hamiltonian if and only if m = n and m, n ≥ 2.

3. Consider a formula Væ3y3zVuE(x, y) ^ E(x, z) ^ (u = x V ¬E(u, y) V ¬E(u, z)). Here, 2, y, z, u e V for some graph G = (V, E), and E() is the edge relation. Assume that Vu¬E(v, v), and that |V| = n. (a) What is the maximal possible number of edges in an undirected graph satisfying this formula? (b) What is the minimal possible number of edges in an directed graph satisfying this formula? Show both that there exists a graph with that many edges (you can just describe how to construct such a graph), and that this number is optimal.

5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).

Please write it in detail.

Let P(Z) = 0.42, P(Y) = 0.39, and P(ZnY) = 0.16. Use a Venn diagram to find (a) P(Z'nY'), (b) P (Z'UY'), (c) P(Z'UY), and (d) P (ZnY'). (a) P(Z'nY') (b) P (Z'UY') (c) P (Z'UY) = (d) P (ZnY') = = = (Type an integer or a decimal.) (Type an integer or a decimal.) (Type an integer or a decimal.) (Type an integer or a decimal.) .. Z Y CO

find the umblic points of the monkey saddle M:z = x3-3xy2.