Let r be a natural number. An r-regular graph is a graph in which all vertices have degree r. Assume that r is odd. Prove that if there exists an r-regular graph with n vertices, then n is even and n >= r+1.

I tried asking this in seperate questions but no body answered . b) Prove that G = K2,12 is planar by drawing G without any edge crossings. (c) Give an example of a graph G whose chromatic number is 3, but that contains no K3 as a subgraph. You must prove that your graph actually has chromatic number 3.

prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

Use the formula deg(v) = 2|E(G)| to find the number of edges of the following vЄV (G) graphs. Classify (count) the vertices by number of neighbors. (a) V(G) = [100]. Edges: for all n and m in [100], n ‡ m, n is adjacent to m if and only if |nm| ≤ 4. (b) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d), (a, b) is adjacent to (c,d) if and only if a = c or b = d. (c) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d), (a, b) is adjacent to (c,d) if and only if |ac| + |bd| = 1. (d) V (G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) is adjacent to (c,d) if and only if |a - c + b-d ≤ 2. (a, b) ‡ (c,d),