Let G be a graph. Prove that G is Eulerian if and only if G has an orientation D where D is an Eulerian digraph.
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- Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets of vertices U,W whose union is V where any path of G from vertices of U to vertices of W contains e.Which of the following graphs contain u,v, and u*v?Let G be a connected graph. Prove that G is Eulerian if and only if each of the blocks of G is Eulerian.
- Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.Prove that if v0 and v1 are distinct vertices of a graph G = (V,E) and a path exists in G from v0 to v1 , then there is a simple path in G from v0 to v1 .Let G be a connected graph of order n and size n. Prove that G contains a single cycle.
- O Prove that for any two simple connected joint graphs G and H, L(G) UL(H) L(GUH).The symmetric difference graph of two graphs G1 = (V, E1) and G2 = (V, E2) on the samevertex set is defined as G1△G2 := (V, E1△E2). Remember that E1△E2 := (E1 \ E2) ∪(E2 \ E1). If G1 and G2 are eulerian, show that every vertex in G1△G2 has even degree(6) Let G be a graph such that for any u, v € V(G), there exists a unique (u, v)-path. Prove that G is a tree.
- 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? ...1prove that the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are.Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G)\ {e}. Then e is a bridge of G if H has a greater number of connected components than G. Assume that G is connected and that e is a bridge of G with endpoints u and v. Show that H has exactly two connected components H₁ and H₂ with u € V (H₁) and v € V(H₂). To this end, you may want to consider an arbitrary vertex w ¤ V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H.